RESEARCH CENTRE

Inria Saclay Centre

2023
ACTIVITY REPORT

Project-Team

MATHEXP

Computer algebra, experimental
mathematics, and interactions

DOMAIN

Algorithmics, Programming, Software and
Architecture

THEME

Algorithmics, Computer Algebra and
Cryptology

Contents

Project-Team MATHEXP

1 Team members, visitors, external collaborators

2 Overall objectives

2.1 Experimental mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Foundations of computer algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

3
3
3
4
5

3 Research program

5
5
3.1 Algebraic algorithms for multivariate systems of equations . . . . . . . . . . . . . . . . . . .
5
3.1.1 Holonomic systems of linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.1.2 Desingularization of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.1.3 Well-foundedness of divide-and-conquer recurrence systems
. . . . . . . . . . . . .
7
3.1.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2 Symbolic integration with parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.2.1 Integrals with boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.2.2 Reduction-based creative telescoping . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Holonomic moment methods
3.2.4 New aspects of symbolic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.5 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Computerized classification of functions and numbers . . . . . . . . . . . . . . . . . . . . . . 10
3.3.1 Practical tests of algebraicity and transcendence for holonomic functions . . . . . . 10
3.3.2 Algorithmic determination of algebraic values of E-functions . . . . . . . . . . . . . . 11
3.3.3 Rational solution of Ricatti-like Mahler equation and hypertranscendence . . . . . . 11
3.3.4 Algorithmic determination of algebraic values of Mahler functions
. . . . . . . . . . 12
3.3.5 Efficient resolution of functional equations with 1 catalytic variable . . . . . . . . . . 12
3.3.6 Classification of solutions for functional equations with 2 catalytic variables . . . . . 12
3.3.7 Deciding integrality of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.8 Algorithmic resolution of Padé-approximation problems
. . . . . . . . . . . . . . . . 13
3.3.9 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Guess-and-prove . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4.1 Univariate guessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4.2 Multivariate guessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Seminumerical methods in computer algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5.1 Seminumerical algorithms for linear differential equations . . . . . . . . . . . . . . . 16
3.5.2 Period computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.3 Scattering amplitudes in quantum field theory . . . . . . . . . . . . . . . . . . . . . . 17
3.5.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Application domains

18

5 Highlights of the year

18
5.1 A whole French year 2023 dedicated to computer algebra . . . . . . . . . . . . . . . . . . . . 18

6 New results

18

6.1 Effective algebraicity for solutions of systems of functional equations with one catalytic

variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.2 Fast algorithms for discrete differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.3 Systems of discrete differential equations, constructive algebraicity of the solutions
. . . . 19
6.4 Reflected Brownian motion in a non convex cone . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.5 A Markovian analysis of IEEE 802.11 broadcast transmission networks with buffering and

back-off stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.6 On the representability of sequences as constant terms
. . . . . . . . . . . . . . . . . . . . . 20
6.7 An algorithmic approach to Rupert’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.8 Algebraicity of hypergeometric functions with arbitrary parameters . . . . . . . . . . . . . . 20
6.9 Fast computation of the N -th term of a q-holonomic sequence and applications . . . . . . 21
6.10 A short proof of a non-vanishing result by Conca, Krattenthaler and Watanabe . . . . . . . . 21
6.11 Persistence probabilities and Mallows-Riordan polynomials
. . . . . . . . . . . . . . . . . . 21
6.12 Continued fractions, orthogonal polynomials and Dirichlet series . . . . . . . . . . . . . . . 21
6.13 Minimization of differential equations and algebraic values of E -functions . . . . . . . . . . 21
6.14 A sharper multivariate Christol’s theorem with applications to diagonals and Hadamard

products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.15 Algebraic solutions of linear differential equations: an arithmetic approach . . . . . . . . . 22
6.16 Refined product formulas for Tamari intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.17 Beating binary powering for polynomial matrices . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.18 On the q-analogue of Pólya’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.19 Positivity certificates for linear recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.20 A signature-based algorithm for computing the nondegenerate locus of a polynomial system 23
6.21 Algorithms for minimal Picard-Fuchs operators of Feynman integrals . . . . . . . . . . . . . 23
6.22 Factoring differential operators on algebraic curves in positive characteristic . . . . . . . . 23
6.23 Reduction-based creative telescoping for definite summation of D-finite functions . . . . . 24
6.24 p-adic algorithm for bivariate Gröbner bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.25 Algorithms in intersection theory in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.26 Motivic geometry of 2-loop Feynman integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.27 Estimating major merger rates and spin parameters ab initio via the clustering of critical

events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.28 Effective homology and periods of complex projective hypersurfaces . . . . . . . . . . . . . 25
6.29 A direttissimo algorithm for equidimensional decomposition . . . . . . . . . . . . . . . . . . 25
6.30 Integer sequences, differential operators and algebraic power series . . . . . . . . . . . . . . 25

7 Partnerships and cooperations

26
7.1 International initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.1.1 Participation in other International Programs . . . . . . . . . . . . . . . . . . . . . . . 26
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.2.1 Horizon Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.3 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.4 Regional initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.2 European initiatives

8 Dissemination

27
8.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.1.1 Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.1.2 Scientific events: selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8.1.3 Journal
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8.1.4 Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.1.5 Scientific expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.1.6 Research administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.2 Teaching - Supervision - Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.2.1 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8.2.2 Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8.3 Scientific animation of the project-team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9 Scientific production

9.1 Publications of the year
9.2 Cited publications

33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Project MATHEXP

1

Project-Team MATHEXP

Creation of the Project-Team: 2022 April 01

Keywords

Computer sciences and digital sciences

A8.1. – Discrete mathematics, combinatorics

A8.3. – Geometry, Topology

A8.4. – Computer Algebra

A8.5. – Number theory

Other research topics and application domains

B9.5.2. – Mathematics

B9.5.3. – Physics

2

Inria Annual Report 2023

1 Team members, visitors, external collaborators

Research Scientists

• Frédéric Chyzak [Team leader, INRIA, Senior Researcher, HDR]

• Alin Bostan [INRIA, Senior Researcher, HDR]

• Guy Fayolle [Inria, Emeritus]

• Pierre Lairez [INRIA, Researcher]

Post-Doctoral Fellows

• Ricardo Thomas Buring [INRIA, Post-Doctoral Fellow, from Sep 2023]

• Claudia Fevola [INRIA, Post-Doctoral Fellow, from Nov 2023]

• Catherine St-Pierre [INRIA, Post-Doctoral Fellow, from Jun 2023]

PhD Students

• Hadrien Brochet [INRIA]

• Alexandre Goyer [Ministère Education, from Sep 2023]

• Alexandre Goyer [INRIA, until Aug 2023]

• Alexandre Guillemot [INRIA, from Oct 2023]

• Alaa Ibrahim [INRIA]

• Pingchuan Ma [AMSS, from Jun 2023]

• Rafael Mohr [UNIV KAISERSLAUTERN]

• Hadrien Notarantonio [UNIV PARIS SACLAY]

• Raphaël Pagès [UNIV BORDEAUX]

• Eric Pichon-Pharabod [UNIV PARIS SACLAY]

• Sergey Yurkevich [UNIV VIENNE, until Aug 2023]

Interns and Apprentices

• Geoffrey Datchanamourtty [INRIA, Intern, from Feb 2023 until Jul 2023]

• Alexandre Guillemot [INRIA, Intern, from Mar 2023 until Aug 2023]

• Aleksandr Storozhenko [INRIA, Intern, from Jun 2023 until Sep 2023]

Administrative Assistant

• Bahar Carabetta [INRIA]

Visiting Scientist

• Catherine St-Pierre [INRIA, from Feb 2023 until Apr 2023]

Project MATHEXP

3

External Collaborator

• Philippe Dumas [retired]

2 Overall objectives

“Experimental mathematics” is the study of mathematical phenomena by computational means. “Com-
puter algebra” is the art of doing effective and efficient exact mathematics on a computer. The MATHEXP
team develops both themes in parallel, in order to discover and prove new mathematical results, often
out of reach for classical human means. It is our strong belief that modern mathematics will benefit more
and more from computer tools. We ambition to provide mathematical users with appropriate algorithmic
theories and implementations.

Besides the classification by mathematical and methodological axes to be presented in §3, MATHEXP’s
research falls into four interconnected categories, corresponding to four different ways to produce
science. The raison d’être of the team is solving core questions that arise in the practice of experimental
mathematics. Through the experimental mathematics approach, we aim at applications in diverse areas
of mathematics and physics. All rests on computer algebra, in its symbolic and seminumerical aspects.
Lastly, software development is a significant part of our activities, with the aim of enabling cutting-edge
applications and disseminating our tools. Each of these four levels is reflected in the thematic axes of the
research program.

2.1 Experimental mathematics

In science, observation and experiment play an important role in formulating hypotheses. In mathemat-
ics, this role is shadowed by the primacy of deductive proofs, which turn hypotheses into theorems, but it
is no less important. The art of looking for patterns, of gathering computational evidence in support of
mathematical assertions, lies at the heart of experimental mathematics, promoted by Euler, Gauss and
Ramanujan. These prominent mathematicians spent much of their time doing computations in order
to refine their intuitions and to explore new territories before inventing new theories. Computations
led them to plausible conjectures, by an approach similar to those used in natural sciences. Nowadays,
experimental mathematics has become a full-fledged field, with prominent promoters like Bailey and
Borwein. In their words [54], experimental mathematics is “the methodology of doing mathematics that
includes the use of computation for

• gaining insight and intuition,

• discovering new patterns and relationships,

• using graphical displays to suggest underlying mathematical principles,

• testing and especially falsifying conjectures,

• exploring a possible result to see if it is worth formal proof,

• suggesting approaches for formal proof,

• replacing lengthy hand derivations with computer-based derivations,

• confirming analytically derived results.”

2.2 Foundations of computer algebra

At a fundamental level, we manipulate several kinds of algebraic objects that are characteristic of com-
puter algebra: arbitrary-precision numbers (big integers and big floating-point numbers, typically with
dozens of thousands of digits), polynomials, matrices, differential and recurrence operators. The first
three items form the common ground of computer algebra [99]. We benefit from years of research on
them and from broadly used efficient software: general-purpose computer-algebra systems like Maple,
Magma, Mathematica, Sage, Singular; and also special-purpose libraries like Arb, Fgb, Flint, Msolve,

4

Inria Annual Report 2023

NTL. Current developments, whether software implementation, algorithm design or new complexity
analyses, directly impact us. The fourth kind of algebraic objects, differential and recurrence operators, is
more specific to our research and we concentrate our efforts on it. There, we try to understand the basic
operations in terms of computational complexity. Complexity is also our guide when we recombine basic
operations into elaborate algorithms. In the end, we want fast implementations of efficient algorithms.

Here are some of the typical questions we are interested in:

• Do some of the solutions of a linear ordinary differential equation (ODE) satisfy a simpler ODE?

This relates to the problem of factoring differential operators.

• Is a given linear partial differential equation (PDE) a consequence of a set of other PDEs? This

relates to the problem of computing Gröbner bases in a differential setting.

• Given a solution f (x, y) of a system of linear PDEs, how to compute differential equations for f (x, 0)

or (cid:82) 1

0 f (x, y)dy? This falls into the realm of symbolic integration questions.

• Given a linear ODE with initial condition at 0, how to evaluate numerically the unique solution at 1

with thousands of digits of precision? This is the gist of our seminumerical methods.

2.3 Applications

Getting involved in applications is both an objective and a methodology. The applications shape the tools
that we design and foster their dissemination.

Combinatorics is a longstanding application of computer algebra, and conversely, computer algebra
has a deep impact on the field. The study of random walks in lattices, first motivated by statistical physics
and queueing theory, features prominent examples of experimental mathematics and computer-assisted
proofs. Our main collaborators in combinatorics are Mireille Bousquet-Mélou (Université de Bordeaux),
Stephen Melczer (University of Waterloo) and Kilian Raschel (Université d’Angers).

Probability theory. Apart from the already mentioned interest in random walks, which is a classical
topic in probability theory, and on which we have an expert, Guy Fayolle, in our group, the main applica-
tions we have in mind are to integrals arising from: 2D fluctuation theory (generalizing arc-sine laws in
1D); moments of the quadrant occupation time for the planar Brownian motion; persistence probability
theory (survival functions of first passage time for real stochastic processes); volumes of structured
families of polytopes also arising in polyhedral geometry and combinatorics. Our main interactions on
these topics are with Gerold Alsmeyer (U. Münster), Dan Betea (KU Leuven), and Thomas Simon (U.
Lille).

Number theory, and especially diophantine approximation, are also fields with longstanding users of

computer algebra tools. For example, the recently discovered sequence of integrals

(cid:90) 4+2i

4−2i

(x − 4 + 2i )4n(x − 4 − 2i )4n(x − 5)4n(x − 6 + 2i )4n(x − 6 − 2i )4n
x6n+1(x − 10)6n+1

dx, n ≥ 0,

whose analysis leads to the best known measure of irrationality of π, can hardly be found by hand [152]. Yet,
the discovery and the proof of such a result requires sophisticated tools from experimental mathematics.
Our main collaborators in number theory are Boris Adamczewski (Université Lyon 1), Xavier Caruso
(Université de Bordeaux), Stéphane Fischler (Université Paris Saclay), Tanguy Rivoal (Université Grenoble
Alpes), Wadim Zudilin (University Nijmegen). Mahler equations are other aspects of number theory, in
relation to automata theory, and appear in several of our research axes. Philippe Dumas, in our group,
and Boris Adamczewski, already mentioned, have long been experts in this topic.

In algebraic geometry, in spite of tremendous theoretical achievements, it is a challenge to apply
general theories to specific examples. We focus on putting into practice transcendental methods through
symbolic integration and seminumerical methods. Our main collaborators are Emre Sertöz (Max Planck
Institute for Mathematics) and Duco van Straten (Gutenberg University).

In statistical physics, the Ising model, and its generalization, the Potts model, are classical in the study
of phase transitions. Although the Ising model with no magnetic field is one of the most important
exactly solved models in statistical mechanics (Onsager won the Nobel prize 1968 for this), its magnetic
susceptibility continues to be an unsolved aspect of the model. In absence of an exact closed form, the

Project MATHEXP

5

susceptibility is approached analytically, via the singularities of some multiple integrals with parameters.
Experimental mathematics is a key tool in their study. Our main collaborators are Jean-Marie Maillard
(SU, LPTMC) and Tony Guttmann (U. Melbourne).

In quantum mechanics, turning theories into predictions requires the computation of Feynman
integrals. For example, the reference values of experiments carried out in particle accelerators are ob-
tained in this way. The analysis of the structure of Feynman integrals benefits from tools in experimental
mathematics. Our main collaborator in this field is Pierre Vanhove (CEA, IPhT).

2.4 Software

We ambition to provide efficient software libraries that perform the core tasks that we need in experi-
mental mathematics. We target especially four tasks of general interest: algebraic algorithms for ma-
nipulating systems of linear PDEs, univariate and multivariate guessing, symbolic integration, and
seminumerical integration.

For several reasons, we want to stay away from a development model that is too tied to commercial
computer algebra systems. Firstly, they restrict dissemination and interoperability. Secondly, they do
not offer the level of control that we need to implement these foundations efficiently. Concretely, we will
develop open-source libraries in C++ for the most fundamental tasks in our research area. Computer
algebra systems, like Sagemath or Maple, are good at coordinating primitive algorithms, but too high-level
to implement them efficiently. We seek solid software foundations that provide the primitive algorithms
that we need. This is necessary to implement the new higher-level algorithms that we design, but also to
reach a performance level that enables new applications. Still, we will strive to expose our libraries to the
prominent computer-algebra systems, especially Maple and Sagemath, used by many colleagues.

Besides, there is a growing interest in the programming language Julia for computer algebra, as shown
by the Oscar project. We already internally use Julia and occasionally some of the libraries Oscar is build
upon, and we want to promote this young ecosystem. It is very attractive to contribute to it, but on the
flip side of the coin, it is too young to offer the same usability as Maple, or even Sagemath. So there is an
assumed element of risk taking in our intent to also make our libraries available to Julia.

3 Research program

3.1 Algebraic algorithms for multivariate systems of equations

At large, MATHEXP deals with algebraic and seminumerical methods. This part goes through the fun-
damental aspects of the algebraic side. As opposed to numerical analysis where numerical evaluations
underlie the basic algorithms, algebraic methods manipulate functions through functional equations.
Depending on the context, different kinds of functional equations are appropriate. Algebraic functions
are handled through polynomial equations and the classical theory of polynomial systems. To deal with
integrals, systems of linear partial differential equations (PDEs) are appropriate. In combinatorics and
number theory appears the need for non-linear ordinary differential equations (ODEs). We also consider
other kinds of functional equations more related to discrete structures, namely linear recurrence relations,
q-analogues and Mahler equations.

The various types of functional equations raise similar questions: is a given equation consequence of
a set of other equations? What are the solutions of a certain type (polynomial, rational, power series, etc.)?
What is the local behavior of the solutions? Algorithms to solve these problems support an important
part of our research activity.

3.1.1 Holonomic systems of linear PDEs

One of the major data structure that we consider are systems of linear PDEs with polynomial coefficients.
A system that has a finite dimensional solution space is called holonomic and a function that is solution
of a holonomic system is called holonomic too. The theory of holonomy is important because it allows
for an algebraic theory of analysis and integration (on this aspect see also §3.2). The basic objects of
holonomy theory are linear differential operators, that are some sort of quasicommutative polynomials,
and ideals in rings of linear differential operators, called Weyl algebras. In this aspect, holonomy theory is

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Inria Annual Report 2023

analogue to the theory of polynomial systems, where the basic objects are commutative polynomials and
ideals in polynomial rings. Some of the important concepts, for example the concept of Gröbner basis,
are also similar. Gröbner bases are a way to describe all the consequences of a set of equations.

As much as Gröbner bases in polynomial rings are the backbone of effective commutative algebra,
Gröbner bases in Weyl algebras of differential operators are the backbone of effective holonomy theory,
which includes integration. In a commutative setting, there has been a long way from the early work
of Buchberger to today’s state-of-the-art polynomial system solving libraries [46]. We will develop a
similar enterprise in the noncommutative setting of Weyl algebras. It will unlock a lot of applications of
holonomy theory.

Following the commutative case, progress in a differential context will come from an appropriate
theory and efficient data structures. We will first develop a matrix approach to handle simultaneous
reduction of differential operators as the F4 algorithm for the polynomial case [92]. The real challenge
here is more practical than theoretical. It is not difficult to come with some F4 algorithm in the differential
case. But will it be efficient? From the experience of modern Gröbner engines in the commutative case, we
know that efficient implementation of simultaneous reduction requires a significant amount of low-level
programming to deal with sparse matrices with a special structure. We also know that many choices,
irrelevant to the mathematical theory, strongly influence the running times. The noncommutativity
of differential operators adds extra complications, whose consequences are still to be understood at
this level. We want to reuse, as much as possible, the specialized linear algebra libraries that have been
developed in the polynomial context [65, 46], but we may have to elude the densification of products
induced by noncommutativity.

On a more theoretical aspect, one step further in the analysis is that the possible analogues of the F5
algorithm [93] are not fully explored in a differential setting. We may expect not only faster algorithms,
but also new algorithms for operating on holonomic functions (Weyl closure for example, see §3.1.2).
Rafael Mohr started a PhD thesis in the team on using F5 for computing equidimensional decompositions
in the commutative case.

3.1.2 Desingularization of PDEs

Among the structural properties of systems of linear differential or difference equations with polynomial
coefficients, the question of understanding and simplifying their singularity structure pops up regularly.
Indeed, an equation or a system of equations may exhibit singularities that no solution have, which
are then called apparent singularities. Desingularization is a process of simplifying a ∂-finite system by
getting rid of its apparent singularities. This is done at the cost of increasing the order of equations, thus,
the dimension of their solution space. The univariate setting has been well studied over time, including in
computer algebra for its computational aspects [34, 33]. This led to the notion of order-degree curve [71,
72, 70]: a given function can cancel an ODE or ORE (ordinary recurrence equation) of small order with a
certain coefficient degree, and also other ODEs or OREs of higher orders, possibly with smaller coefficient
degrees. In certain applications, the ODE or ORE of minimal order may be too large to be obtained by
direct calculations. It appears that the total size of the equations, that is, the product of order by degree,
can be more relevant to optimize the speed of algorithms. This is a phenomenon that we observed first in
relation to algebraic series [59], and we want to promote further this idea of trading minimality of order
for minimality of total size, with the goal of improved speed. On the other hand, apparent singularities
have been defined only recently in the multivariate holonomic case [73].

Our project includes developing good notions and fast heuristic methods for the desingularization
of a ∂-finite system, first in the differential case, where it is expected to be easier, then in the case of
recurrence operators.

Moreover, fast algorithms will be obtained for testing the separability of special functions: in a
nutshell, this problem is to decide whether the solutions to a given system also satisfy linear differential
or difference equations in a single variable, and algorithmically this corresponds to obtaining structured
multiples of operators with a structure similar to that for desingularization.

In the multivariate case, the operation of saturating an ideal in the Weyl algebra by factoring out
(and removing) all polynomial factors on the left is known under the name of Weyl closure. This relates
to desingularization as the Weyl closure of an ideal contains all desingularized operators. Weyl closure
also is a relative of the radical of an ideal in commutative algebra: given an ideal of linear differential

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operators, its Weyl closure is the (larger) ideal of all operators that annihilate any function solution to the
initial ideal. Computing Weyl closure applies to symbolic integration, and algorithms exist to compute it
[149, 148], although they are slow in practice. Weyl closure also plays an important role in applications
to the theory of special functions, e.g., in the study of GKZ-systems (a.k.a. A-hypergeometric systems)
[124], and in relation to Fischer distribution and maximum likelihood estimation in statistics [37, 100].
Algorithms for Weyl closure should then be obtained, by basing on desingularization as a subtask.

3.1.3 Well-foundedness of divide-and-conquer recurrence systems

Converting a linear Mahler equation with polynomial coefficients (see §3.3.3) into a constraint on the
coefficient sequence of its series solutions results in a recurrence between coefficients indexed with
rational numbers, which must be interpreted to be zero at noninteger indices. The recurrence can be
replaced with a system of recurrences by cases depending on residues modulo some power of the base b.
The literature also alternatively introduces recurrences with indices expressed with floor/ceiling functions,
typically so for fine complexity analysis of divide-and-conquer algorithms. For sequences that can be
recognized by automata (“automatic sequences”) and their generalizations (“b-regular sequences”), it is
natural to consider a system of recurrences on several sequences, with a property of closure under certain
operations of taking subsequences: restricting to even indices, or odd indices, or more generally indices
with a given residue modulo the base b. This variety of representations calls for algorithms to be able
to convert from one another, to check the consistency of a given system of recurrences, and to identify
those terms of the sequence that determine all others (which are typically not just a few first terms). In
the continuation of [77] that developed a Gröbner-bases theory as a pre-requisite for this goal, we will
address those problems of conversion and well-foundedness.

3.1.4 Software

Software development is a real challenge, regarding the symbolic manipulation of linear PDEs. While
symbolic integration has gained more and more recognition, its execution is still reserved to experts.
Providing a highly efficient software library with functionalities that come as close as possible to the
actual integrals, rather than some idealized form, will foster adoption and applications. In the past, the
lack of solid software foundations has been an obstacle in implementing newly developed algorithms
and in disseminating our work. It was the case, for example, for our work on binomial sums [61], or the
computation of volumes [118], where having to use an integration algorithm implemented in Magma has
been a major obstacle.

What is lacking is a complete tool chain integrating the following three layers:

1. the computation of Gröbner bases of holononomic systems, as discussed in §3.1.1;

2. the basic algorithms for manipulating holonomic systems, such as the desingularization discussed

in §3.1.2 but also the classical aspects of symbolic integration;

3. the algorithms relevant for applications, including all the aspects covered in §3.2.

The first layer of the toolchain will be developed in C++ for performance but also to open the way to an
integration in free computer algebra systems, like Sagemath or Macaulay2. We will benefit from years
of experience of the community and close colleagues in implementing Gröbner basis algorithms in the
commutative case. The third layer of the toolchain should be easily accessible for the users, so at least
available in Sagemath. Some of our current software development, related to the second layer, already
happens in Julia (as part of R. Mohr’s PhD work).

3.2 Symbolic integration with parameters

Among common operations on functions, integration is the most delicate. For example, differentiation
transforms functions of a certain kind into functions of the same kind; integration does not. For this
reason, integration is also expressive: it is an essential tool for defining new functions or solving equations,
not to mention the ubiquitous Fourier transform and its cousins. Integration is the fundamental reason
why holonomic functions are so important: integrals of holonomic functions are holonomic. Algorithms

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to perform this operation enable many applications, including: various kinds of coefficient extractions
in combinatorics, families of parametrized integrals in mathematical physics, proofs of irrationality in
number theory, and computations of moments in optimization.

Given a function F (t, x) of two blocks of variables t = t1, . . . , ts and x = x1, . . . , xn, and an integration

domain Ω(t) ⊆ Rn, how to compute the function

(cid:90)

G(t) =

F (t, x)dx?

Ω(t)

Concretely, F (t, x) is described by a system of linear PDEs with polynomial coefficients, Ω(t) is given
by polynomial inequalities, and we want a system of PDEs describing G(t). Note here the presence of
parameters which makes it possible to describe the result of integration with PDEs. When there are no
parameters, the result is a numerical constant. Even though we deal with them in an entirely different
way (see §3.5), we still mostly rely on symbolic integration with parameters.

From the algebraic and computational point of view, integration has several analogues. Discrete sums
are the prominent example, but there are also q-analogues, Mahlerian functions, and some others. At
large, algorithms for symbolic integration, or its analogues, perform a sort of elimination in a ring of
differential operators. There are some links with elimination theory and related algorithms as developed
for the study of polynomial systems of equations.

Symbolic integration is an historical focus of MATHEXP’s founding members with many significant
contributions. Compared to our previous activities, we want to put more emphasis on software develop-
ment. We are at a point where the theory is well understood but the lack of efficient implementations
hinders many applications. Naturally, this effort will rest on the results obtained in §3.1.

3.2.1 Integrals with boundaries

The algebraic aspects of symbolic integration are best understood when the integration domain has no
boundary: typically Rn or a topological cycle in Cn. Indeed, in this context we have the so-called telescopic
relation which states that the integral of a derivative vanishes: for example, if H (t, x) is rapidly decreasing,
then

(cid:90)

Rn

∂H
∂xi

dx = 0.

It gives a nice algebraic flavor to the problem of symbolic integration and reduces it to the study of the
quotient space F /(cid:161) ∂
F (cid:162), where F is a suitable function space containing the integrand. A
∂x1
large part of the algorithms developed so far focuses on this case. Yet, many applications do not fit in this
idealized setting. For example, Beukers’ proof of the irrationality of ζ(3) [49] uses the two integrals

F + · · · + ∂
∂xn

(cid:212)

γ R dxdydz and (cid:208)

[0,1]3 R dxdydz, where R(t , x, y, z) =

1
1−(1−x y)z−t x y z(1−x)(1−y)(1−z) .

The first one, where the integration domain is some complex cycle γ, is well handled by current algorithms.
The second is not, and this is unsatisfactory for further applications of symbolic integration in number
theory. In this particular case, we may think of an algorithm that would reduce the integration on the
cube to an integration without boundary and an integration on the boundary of the cube. This boundary
just consists of 6 squares, which calls for a recursive procedure. Unfortunately, the integration domain
touches the poles of the integrand, so operations like integrating only part of a function or integration by
parts or differentiation under the integral sign may not be meaningful by lack of integrability. It is not
known how to deal with this issue automatically. For more general domains of integration, it is not even
clear what kind of recursive procedure can be applied.

The next generation of symbolic integration algorithms must deal with integrals defined on domains
with boundaries. The framework of algebraic D-modules seems to be very appropriate and already
features some algorithms. But this is not the end of the story, as this line of research has not led yet
to efficient implementations. We identified two ways of action to reach this goal. Firstly, existing
algorithms [130, 131] put too much emphasis on computing a minimal-order equation for the integral.
While this is an interesting property, other kinds of integration algorithms have successfully relaxed this
condition. For example, for integrating rational functions, the state-of-the-art algorithm [117] depends
on a parameter r > 0. The computed equation is minimal only for r large enough, which corresponds to

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the degeneration rank of some spectral sequence [86]. In practice, this has never been an obstacle: most
of the time we obtain a minimal equation with a small value of r . For the few remaining cases, we will
soon propose a generalized procedure to minimize the equation a posteriori; this will be a consequence
of a work on univariate guessing (see §3.4.1) that bases and expands on [62]. The algorithm by small
values of r applicable in most cases already outperforms previous ones in terms of computational
complexity [60] and practical performance, being able to compute integrals that were previously out of
reach. We consider it to be a special case of the general algorithm that we want to develop, and a proof of
feasibility. However, the effort will be vain without significant progress on the computation of Gröbner
bases in Weyl algebras. Fortunately, and this is the second way of action, we think that the framework
of algebraic D-modules enables efficient data structures modeled on recent progress in the context of
polynomial systems. Progress in this direction (as explained in §3.1.1) will immediately lead to significant
improvement for symbolic integration.

3.2.2 Reduction-based creative telescoping

The approach to symbolic integration based on creative telescoping is a definite expertise of the team.
Although the approach is difficult to use for integrals with boundaries, it still has many appeals. In
particular, it generalizes well to discrete analogues. Recently, the team has initiated the development of a
new line of algorithms, called reduction-based. After continuing work, this line has not yet been extended
to full generality [58, 109]. These recent theoretical developments are not yet reflected in current software
packages (only prototype implementations exist) and therefore their practical applicability, and how the
algorithms compare, is not yet fully understood. Filling these gaps will be a good starting point for us,
but the ultimate goal will be to formulate analogue algorithms for the difference case (summation of
holonomic sequences), for the q-case, and for general mixed cases. We expect that these advances in the
theory will have a great impact on various applications.

3.2.3 Holonomic moment methods

In applied mathematics, the method of moments provides a computational approach to several import-
ant problems involving polynomial functions and polynomial constraints: polynomial optimization,
volume estimation, computation of Nash equilibria, . . . [121]. This method considers infinite-dimensional
linear optimization problems over the space of Borel measures on some space Rn. They admit finite-
dimensional relaxations in terms of linear matrix inequalities where a measure µ is represented approx-
imately by a finite number of moments (cid:82) x
α

d µ.

From the holonomic point of view, the generating function of the moments of µ — or, equivalently,
the characteristic function φ(u) = (cid:82) exp(i u · x)d µ — is holonomic for a large class of measures µ (which
includes all measures that appear in current applications of the method of moments). This remark
already unlocks some applications where the current bottleneck is the computation of many moments:
differential equations on φ(u) reflect recurrence relations on the moments, and computing the former
with symbolic integration will lead to efficient algorithms for computing the moments.

A line of research developed recently [112, 122, 120, 144, 123] focuses on reducing the size of the
matrices in the linear matrix inequalities (LMI) involved in the relations by using pushforward measures.
For example, let us consider a polynomial f ∈ R[x1, . . . , xn] and the problem of computing the volume
of (cid:169)p ∈ [0, 1]n (cid:175)
(cid:175) f (p) ≥ 0(cid:170). The article [106] solves this problem with a linear program over Borel measures
on [0, 1]n. Using the pushforward measure, the work [112] reduces to a linear program over measures
on R, supposedly much easier to solve. However, this comes at the cost of computing the moments µk =
(cid:82)
[0,1]n f k d x1 · · · d xn for increasingly large values of k. While this is an elementary task (it is enough to
expand f k ), the number of monomials to compute is ∼ 1
n! (k deg( f ))n, for large k, and this becomes the
bottleneck of the method. The computation of the generating series (cid:80)
µk t k using symbolic integration
enables the computation of a linear recurrence relation, of size deg( f )n at most, for the moments µk , and
we can compute the µ0, . . . , µk in O(k deg( f )n) arithmetic operations only, or ˜O(k2 deg( f )n) bit operations.
This should be a low-hanging fruit as soon as we have reasonable implementations of symbolic integration
on domains with boundaries (see §3.2.1). Naturally, the constant in the big O hides the size of the ODE of
which the generating function is solution, and it may be exponential in n. But this is only a worst-case
bound, and any nongeneric geometric property will tend to make this ODE smaller.

k≥0

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k≥0

One step further, we will try to interpret the whole moment method in the holonomic setting. The
differential equation for (cid:80)
µk t k not only enables the computation of the moments µk , it somehow
encodes all the µk . Recovering numerical values, such as the volume, from this differential equation
is akin to the seminumerical algorihtms we know (see §3.5.1). As a next step, we will study how some
optimization problems treated by the method of moments behave in this holonomic setting. We think
especially of the problems of chance optimization and chance constrained optimization [111]: in the
former, one maximizes the probability of success over the design parameters; in the latter one optimizes
a goal while ensuring that some probability remains low.

3.2.4 New aspects of symbolic integration

Mahlerian telescopers. Here the aim is to determine the relations satisfied by a solution of a Mahler
equation (see §3.3.3). A natural generalization is to search for relations among solutions of different
Mahler equations. Our objective is to provide an algorithmic answer to this generalization, for (Laurent)
power series yi solutions of inhomogeneous first-order equations of the form yi (z p ) + ai (z)yi (z) =
bi (z), with coefficients in ¯Q(z). We will start with the easy case where all ai ’s are equal to 1. Under
this assumption, a theorem of Hardouin and Singer guarantees that there exists an algebraic relation
with coefficients in ¯Q(z) between y1, . . . , yn if and only if there exists a Mahlerian telescoper between
the bi (z). (This originates in Hardouin’s PhD thesis and was generalized in [104].) We will work on making
algorithmic such an existence test, and if possible the calculation of such telescopers. For this, we will be
inspired by existing algorithms for calculating telescopers for other types of functional operators.

D-algebraicity and elliptic telescoping. Random walks confined to the quarter plane is a well studied
topic, as testified by the book [95]. A new algebraic approach, relying on the Galois theory of difference
equations, has been introduced in [88] to determine the nature of the generating series of such walks.
This approach gives access to the D-algebraicity of the generating functions, that is, to the knowledge of
whether they satisfy some differential equations (linear or non-linear). More precisely, D-algebraicity
is shown to be equivalent to the fact that a certain telescopic equation, similar to the one appearing in
the classical context of creative telescoping, but defined on an elliptic curve attached to the walk model,
admits solutions in the function field of that curve. For the moment, the corresponding telescoping
equations are solved by hand, in a quite ad-hoc fashion, using case-by-case treatment. We aim at
developing a systematic and automatized approach for solving this kind of elliptic creative telescoping
problems. To this end, we will import and adapt our algorithmic methods from the classical case to the
elliptic framework.

3.2.5 Software

Because of the dependency of the software pertaining to symbolic integration on developments on
multivariate systems, our goals related to software on symbolic integration have been described in §3.1.4.

3.3 Computerized classification of functions and numbers

Classifying objects, determining their nature, is often the culmination of a mature theory. But even the
best established theories can be impracticable on a concrete instance, either by a lack of effectiveness
or by a computational barrier. In both cases, an algorithm is missing: we have to systematize, but also
effectivize and automate efficiently. This is what we propose to do, in order to solve classification problems
relating to numbers, analytical functions, and combinatorial generating series.

3.3.1 Practical tests of algebraicity and transcendence for holonomic functions

It is an old question addressed by Fuchs in the 1870s of whether one can decide if all solutions of a
given linear differential equation are algebraic. Singer showed in [143] that there exists an algorithm
which takes as input a linear differential equation with coefficients in Q[x], and decides in a finite
number of steps whether or not it has a full basis of algebraic solutions. If the answer is negative, this
does not automatically exclude the possibility that a particular solution is algebraic. (For instance, the

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′

′ = 0 has not only the algebraic solution 1, but also the transcendental
)
linear differential equation (x y
holonomic solution y(x) = log(x).) However, a recent refinement of Singer’s 1979 method can be used
to solve in principle Stanley’s open problem [147, 4(a)]: given a holonomic power series y(x) by an
annihilating linear differential equation and sufficiently many initial terms in its expansion, decide if
y(x) is algebraic or transcendental. Unfortunately, the corresponding algorithm is too slow in practice,
because of its high computational complexity1. An interesting question is to find efficient alternatives
that are able to answer Stanley’s question on concrete difficult examples.

An approach that always works is the algorithmic guess-and-prove paradigm (see §3.4): one guesses a
concrete polynomial witness, and then post-certifies it. This method is very robust and works perfectly
well, but it may fail on examples with minimal polynomial much larger than the input differential
equation. For instance, in an open question by Zagier [151], the input differential equations have order 4,
but the (estimated) algebraic degree of the desired solution is 155520, hence much too large to allow
the computation of the minimal polynomial. (Note that the estimate is obtained using seminumerical
methods evoked in §3.5).

We aim at designing various pragmatic algorithmic methods for proving algebraicity or transcendence
in such difficult cases. First, the algebraic nature of the holonomic function is tested heuristically, using a
mixture of numeric and p-adic methods (e.g., monodromy estimates and p-curvature computations). In
cases where transcendence is conjectured, the method we will develop is an application of the minimiza-
tion algorithms in §3.4.1: after finding a minimal-order ODE, an analysis of singularities is sufficient to
decide transcendence, at least for interesting subclasses of inputs (e.g., a certain class of generating series
of binomial sums). In cases where algebraicity is conjectured, we plan to apply computational strategies
inspired by effective differential Galois theory and effective invariant theory, in particular by the recent
work [41].

3.3.2 Algorithmic determination of algebraic values of E-functions

E -functions are holonomic and entire power series subject to some arithmetic conditions; they generalize
the exponential function. The class contains most of the holonomic exponential generating functions
in combinatorics and many special functions such as the Airy and the Bessel functions. Given an
E -function f represented implicitly by a linear differential equation (and enough initial terms), the
question is to determine algorithmically the algebraic numbers α such that f (α) is algebraic. A recent
article by Adamczewski and Rivoal [36] proves that the problem is decidable. It relies on important
works by Siegel [142], Shidlovskii [141], and Beukers [50]. However, the underlying algorithm has no
practical applicability. We will obtain an improved version of this algorithm, by accelerating its bottleneck,
which consists in computing a linear differential operator of minimal order satisfied by f . This will
take advantage of the results obtained in §3.4.1. By continuing the line of work opened in [62], the idea
is now to exploit the particular structure of differential equations satisfied by E -functions, and to use
bounds produced by calculation on the considered equation rather than theoretical bounds such as
“multiplicity lemmas". Our previous improvements will make this algorithm practical. We will also address
an extension of the theory that also determines cases of algebraic dependency between evaluations of
E -functions [97].

3.3.3 Rational solution of Ricatti-like Mahler equation and hypertranscendence

Mahler equations are functional equations that relate a function f (z) with f (z p ), f (z p2
), etc., for some
integer p > 0. The study of Mahler equations is motivated by Mahler’s work in transcendence, as well as
by the study of automatic sequences, produced by finite automata (see §3.1.3). From a computer algebra
perspective, the basic tasks concerning Mahler equations are poorly understood, compared to differential
or recurrence equations.

1It relies, among other costly operations, on factoring differential operators, which is known to be a highly expensive procedure,
of complexity (N L )O(r 4), where L is the bitsize of the input operator, r its order, and N ≤ exp(L · 2r )2r
[101]. It also relies
on deciding whether a non-linear (Ricatti-type) differential equation of order r − 1 has an algebraic solution of degree at most
M := (49r )r 2
; this step itself relies on deciding non-emptiness of a constructible set defined by polynomials in M variables (and
/y = u has
potentially huge degrees). It also relies on the famously difficult Abel’s problem: given an algebraic function u, decide if y
an algebraic solution.

′

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Roques designed an algorithm for the computation of the Galois group of Mahler equations of
order 2 [139]. This group reflects the algebraic relations between the solutions. So its computation is
relevant in transcendence theory. Roques’ algorithm relies on deciding the existence of rational solutions
to some nonlinear Mahler equations that are analogues of Riccati differential equations. For this task,
Roques proposes an algorithm reminiscent of Petkovšek’s algorithm [134], with an exponential arithmetic
complexity as it has to iterate through all monic factors of well-identified polynomials. Building on recent
progress in the linear case [76], we want to obtain a polynomial-time algorithm for this decidability
problem, or at least one that is not exponentially sensitive to the degree of the polynomial coefficients of
the equation.

An application of this work will be a new algorithm to decide the differential transcendence of
solutions of Mahler equations of order 2, following a criterion given by Dreyfus, Hardouin and Roques
(see [87, 139]). This would make it possible to prove new results about some classical Mahler functions
and the relations between them. An example will be to reprove and extend the hypertranscendence of
the solutions to the Mahler equation satisfied by the generating series of the Stern sequence [85].

3.3.4 Algorithmic determination of algebraic values of Mahler functions

We aim at studying the special values of Mahler functions, going through the search for algebraic values
and more generally for algebraic relations between values. We will resume the analysis of the algorithm
in [35], to highlight its computational limitations, before optimizing its subtasks. We are thinking in
particular of the rationality test, for which an algorithm was given in [43] and another of better complexity
has appeared recently [76], and of the search for minimal equations, for which structured linear algebra
techniques must allow practical efficiency.

3.3.5 Efficient resolution of functional equations with 1 catalytic variable

In enumerative combinatorics, many classes of objects have generating functions that satisfy functional
equations with “catalytic” variables, relating the complete function with the partial functions obtained by
specializing the catalytic variables. For equations with a single catalytic variable, either linear or nonlinear,
solutions are invariably algebraic. This is a consequence of Popescu’s theorem on Artin approximation
with nested conditions [137, Thm. 1.4], a deep result in commutative algebra. However, the proof of
this qualitative result is not constructive. Hence, to go further, towards quantitative results, different
approaches are needed. Bousquet-Mélou and Jehanne proposed in [64] a method which applies in
principle to any equation of the form P (F (t ; x), F1(t ), . . . , Fk (t ), t , x) = 0, where x is a (single) catalytic
variable, that admits a unique solution (F, F1, . . . , Fk ) ∈ Q[x][[t ]] × Q[[t ]]k . The method is based on a
systematic constructive approach, which first derives from the functional equation a (highly structured)
algebraic elimination problem over Q(t ) with 3k unknowns and 3k polynomial equations, whose degree
is linear in the degree δ of the input functional equation. The problem is already nontrivial for k = 1,
but most interesting combinatorial applications require k > 1, and current methods are only able to
tackle functional equations with small values of k (at most 3) and small total degree δ (at most 4). We
will provide unified, systematic and robust algorithms for computing polynomial equations exhibiting
the algebraicity of solutions for functional equations with one catalytic variable, building on [64]. The
ideal goal is to be able to exploit the geometry and symmetries of the elimination problems arising from
the approach in [64]. The final objective is to produce efficient implementations that can be used by
combinatorialists in order to solve their functional equations with one catalytic variable in a click.

3.3.6 Classification of solutions for functional equations with 2 catalytic variables

When several catalytic variables are involved, Popescu’s theorem does not hold anymore. The solutions
are not necessarily algebraic anymore, and even holonomy is not guaranteed, even in the linear case.

In the linear case, our the main objective is to fully automatize the resolution of linear equations with
two catalytic variables coming from lattice walk questions, when the walk model admits Tutte invariants
and decoupling functions. A first nontrivial challenge will be to produce a new computer-assisted proof
of algebraicity for the famous Gessel model, different in spirit from the first proof [57]: instead of guess-
and-proof, we will be inspired by the recent “human” proofs in [63, 45] relying on Tutte invariants. There
are several nontrivial subproblems, both on the mathematical and algorithmic sides. One of them is to

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determine if a model admits invariants and decoupling functions, and if so, to compute them. A first step
in this direction was recently done by Buchacher, Kauers and Pogudin [68], in the simpler case when one
looks for polynomials instead of rational functions.

In the nonlinear case with two catalytic variables, few results exist, and almost no general theory.
These equations occur systematically when counting planar maps equipped with an additional structure,
for instance a colouring (or, a spanning tree, a self-avoiding walk, etc.). On this side, the study will be of a
more prospective nature. However, we envision the resolution of several challenges. A first objective will
be to test various guess-and-prove methods on Tutte’s equation [150] satisfied by the generating function
of properly q-colored triangulations of the sphere. Any kind of progress on it will be an important success–
for instance, proving algebraicity of the solution by a computer-driven approach even for particular
values of q such as q = 2 and q = 3. A second objective will be to automatize the strategy based on Tutte
invariants employed by Bernardi and Bousquet-Mélou, and to solve the more general equation (Potts
model on planar maps) in an automated fashion. This is interesting already for q = 2; in this case, proofs
already exist in [44], but they use various ad-hoc tricks. We aim at solving the conjectures in [44] for q = 3,
concerning the enumeration of properly 3-colored near-cubic maps, by any combination of methods
(guess-and-prove, geometric-driven elimination for structured polynomial systems, Tutte invariants).

3.3.7 Deciding integrality of a sequence

Given enough terms of a sequence, it is possible to reconstruct a linear recurrence relation of which the
sequence is a solution, if there is one. For example, with the nine numbers 1, 3, 13, 63, 321, 1683, 8989,
48639 and 265729, one can reconstruct the recurrence relation (n +1)un −(6n +9)un+1 +(n +2)un+2 = 0 for
(cid:162)(cid:161)n+k
(cid:162),
the Delannoy numbers. We would also like to be able to reconstruct the closed form un = (cid:80)n
k
because it reveals arithmetic information absent from the recurrence, such as the integrality of the
numbers un. The search for a closed form can start by obtaining candidates in a heuristic way, since the
summation algorithms make it possible to rigorously prove or disprove a posteriori that the reconstructed
closed form is indeed correct.

(cid:161)n
k

k=0

3.3.8 Algorithmic resolution of Padé-approximation problems

Most of the proofs of irrationality of some constant c construct a sequence of rational numbers approx-
imating c with a tight control on the growth of the denominator. Typically, c = F (1) for some holonomic
function F , and approximations of F by rational functions may lead to rational approximations of c, by
evaluating at 1. Good candidates for approximating F are the Padé approximants of F , originating in
Hermite’s work [107]. But approximations that actually lead to interesting Diophantine results are rare
gems. More recently, a general course of action has emerged [51, 146, 98] to deal with the case of multiple
zeta values (MZV). It is based on the simultaneous approximation of polylogarithm functions by rational
functions. We are looking to automate this approach and to extend its field of application.

We will use computer-assisted symbolic and numerical computations for the construction of a
relevant Padé-approximation problem. Then, the resolution of the problem must be automated. This is
fundamentally a computational problem in a holonomic setting. The natural approach here is guess-
and-prove: we first guess what could be a closed-form formula for the solution by computing explicitly
the solutions for some fixed values of n, then we prove that the guess indeed leads to a solution (which
must be unique if the original problem is well-posed). The last step will typically use symbolic integration
and Gröbner bases. Similar guess-and-prove approaches in a holonomic setting already gave several
Diophantine results [152] but Padé approximation has not been tackled yet in this way.

3.3.9 Software

Our future algorithm for computing a linear differential equation of minimal order satisfied by a given
holonomic function will be implemented and made available to users. This may include the application
to the determination of algebraic values of E-functions. We will do the same concerning linear Mahler
equations of minimal order satisfied by given Mahler functions, and concerning the determination of their
algebraic values. Our work on solving equations with catalytic variables has started rather recently, so it is
still too early to decide the form that related software should take, but we definitely ambition to provide

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combinatorialists with an implementation that exhibits the algebraic and/or differential equations they
are after.

3.4 Guess-and-prove

Pólya has theorized and popularized a “guess-and-prove” approach to mathematics in remarkable
books [136, 135]. It has now became an essential ingredient in experimental mathematics, whose power
is highly enhanced when used in conjunction with modern computer algebra algorithms. This paradigm
is a key stone in recent spectacular applications in experimental mathematics, such as [57][115, 116]. The
first half (the guessing part) is based on a “functional interpolation” phase, which consists in recovering
equations starting from (truncations of) solutions. The second half (the proving part) is based on fast
manipulations (e.g., resultants and factorization) with exact algebraic objects (e.g., polynomials and
differential operators).

In what follows we mostly focus on the guessing phase. It is called algebraic approximation [66] or
differential approximation [113], depending on the type of equations to be reconstructed. For instance,
differential approximation is an operation to get an ODE likely to be satisfied by a given approximate
series expansion of an unknown function. This kind of reconstruction technique has been used at least
since the 1970s by physicists [103, 102, 110], under the name recurrence relation method, for investigating
critical phenomena and phase transitions in statistical physics. Modern versions are based on subtle
algorithms for Hermite–Padé approximants [42]; efficient differential and algebraic guessing procedures
are implemented in most computer algebra systems.

In the following subsections, we describe improvements that we will work on.

3.4.1 Univariate guessing

Minimization. A first task is to optimize the search for the minimal-order ODE satisfied by a given
holonomic series. Feasibility is already known from the recent [36, §3], but the corresponding algorithm
is not efficient in practice, because it relies on pessimistic degree bounds and on pessimistic multiplicity
estimates. We will design and implement a much more efficient minimization algorithm, which will
combine efficient differential guessing with a dynamic computation of tight degree bounds.

Post-certification.
“Multiplicity lemmas” are theorems concluding that an expression representing
a formal power series is exactly zero under the weaker assumption that the expression is zero when
truncated to some order. In general, the expression is a differential polynomial in a series, but interesting
subcases are non-differential polynomials, to test algebraicity, and linear differential expressions, to
test holonomicity. In good situations, multiplicity lemmas turn guessing into a proving method or
even a decision algorithm. A particularly nice form of a multiplicity lemma is available for polynomial
expressions [55, Ch. 7], and a similar result exists for linear ODEs [48]. We will implement such bounds as
proving procedures, and we will generalize the approach to other kinds of expressions, e.g., expressions
in divided-difference operators that appear in combinatorics, e.g., in map enumeration [64].

Recombination. Generating functions appear in a variety of classes of increasing complexity, in relation
to the equations they satisfy. A third subtask relates to the search for an element in a lower complexity class
inside the solution set of a higher complexity class. For instance, can a linear or some other combination
of non-holonomic series be holonomic? Can a linear combination of holonomic series be algebraic, or
even rational? A promising ongoing result, obtained incidentally in the work on Riccati-type solutions for
Mahler equations (see §3.3.3), performs a similar guessing by a suitable search for constrained Hermite–
Padé approximants after computing the whole module of approximants. But the main expected impact
of the approach would be for differential analogues, and we will strive to generalize the approach, taking
advantage of the formal analogy between many types of linear operators.

Preparing data. As guessing often requires to first prepare a lot of data, developing fast expansion
algorithms for classes of equations is also related to guessing. In this direction, we plan to design a
fast algorithm for the high-order expansion of a DD-finite series (i.e., series satisfying linear differential
equations with holonomic coefficients). The complexity of the homologue problem for a linear ODE

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with series coefficients is quasi-linear in the truncation order; that for a linear ODE with polynomial
coefficients is just linear. For DD-finite series, we plan to interlace the two approaches without first
expanding the series coefficients of the input equation to the wanted order, so as to avoid a large constant
and a logarithmic factor.

3.4.2 Multivariate guessing

Multivariate aspects of guessing relate to activities that we plan to develop as a means of strengthening
scientific collaborations with colleagues in Paris (PolSys, Sorbonne U.) and Linz (Johannes Kepler Univer-
sity Linz, Austria). How soon the research happens will depend on how interaction with those colleagues
evolves.

Trading order for degree. An established technique in the univariate case is known as “trading order for
degree”. It is based on the observation that minimal order operators tend to have very high degree, while
operators of slightly higher order often have much smaller degrees and are therefore easier to guess. A
candidate for the minimal order operator is then obtained as greatest common right divisor of two guessed
operators of nonminimal order. We will extend this successful technique to the multivariate case. The
desired output in this case is a Gröbner basis of a zero-dimensional annihilating ideal. The coefficients of
the Gröbner basis elements are high-degree polynomials, and the idea is, as in the univariate case, not to
guess them directly, but to guess ideal elements of smaller total size and to compute the Gröbner basis of
them. As Gröbner basis computations can be costly, the alternative operators will clearly already have
to be “close” to a Gröbner basis in order for the idea to be beneficial. The questions are: what should
close to a Gröbner basis mean, how close should the operators be chosen, how much degree drop can be
expected then, and how do the answers to these questions depend on the monomial order?

In another direction, we plan to exploit the generalized Hankel structure
Exploiting nested structures.
of the matrices that appear when modeling linear recurrence relations guessing through linear algebra.
Regarding relations with constant coefficients, this finds applications in polynomial system solving
through the spFGLM algorithm [90, 91] for finding a lexicographic Gröbner basis. The linear system is
block-Hankel with blocks sharing the same structure, and this recursive structure has the same depth as
the number of variables. Yet, up to now, only one layer of the structure is handled using fast univariate
polynomial arithmetic, then the other ones are dealt with by noting that the matrix has a quasi-Hankel
structure and using fast algorithms for this type of matrix [56]. However, the displacement rank of this
matrix is not small; hence, not taking into account the full structure of the matrix is suboptimal. This
is related to [47] for computing linear recurrence relations with constant coefficients using polynomial
arithmetic and [128] for computing multivariate Padé approximants. Analogously, the linear system
modeled for guessing linear recurrence relations with polynomial coefficients is highly structured. It is
the concatenation of matrices as above, yet these matrices are not independent, as they are all built from
the same sequence. Even in the univariate case, the Beckermann–Labahn algorithm is not able to exploit
this extra structure in order to be quasi-optimal in the input size. Hence, we would like to investigate how
to do so.

In addition to the structure in the modeling, we want to exploit the structure of the sequences that
come from applications. For instance, in the enumeration of lattice walks, the nonzero terms often lie
in a cone and a lattice, and they are invariant under the action of a finite group. The goal is to take this
structure into account in order to build smaller systems for the guessing, and to avoid the generation of
more sequence terms than necessary.

3.4.3 Software

We will implement fast algorithms for computing Hermite–Padé approximants of various types [42].
This will include modular integers, integers (via modular reconstruction), simple approximants, and
simultaneous approximants. With such a fast, robust implementation at hand, we will also be able to
address the guessing of algebraic differential equations (ADE), going beyond the linear case. Our use
of state-of-the-art algorithms for computing approximants (including the “superfast” one) will ensure
that we outperform earlier implementations such as Guess (by Hebisch and Rubey) and GuessFunc

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(by Pantone). We will also develop a variant of trading order for degree for the nonlinear setting. Our
implementation will automate the critical selection of derivatives, powers, and coefficient degrees needed
to reconstruct an ADE.

3.5 Seminumerical methods in computer algebra

The methods in this research axis deal directly with numbers but, following Knuth [114], they are properly
called seminumerical because they lie on the borderline between symbolic and numeric computations.
While numerical methods process numerical data and generate further numerical data, our seminumer-
ical methods process exact data, generate high-precision numerical data and reconstruct exact data.
In this perspective, the basic unit is not the IEEE-754 floating-point number, but arbitrary precision
numbers, typically several thousand decimal places, sometimes more. The crux is not numerical stability,
but computational complexity as the number of significant digits goes to infinity. When a number is
known at such a high precision, it reveals fundamental structures: rationality, algebraicity, relations with
other constants, etc. High-precision computation is a recurring useful tool in the field of experimental
mathematics [39]. In some situations, it enables a guess-and-prove approach. In some others, we are
unable to step from “guess” to “prove” but overwhelming numerical evidence is enough to shape a convic-
tion. A celebrated example is the experimental discovery of the BBP formula for π [40] (that was proved
after its initial guessing). More recently, all the conjectures (some of which became theorems) about
multiple zeta values, a hot topic in number theory and mathematical physics, start from high-precision
numerical data.

3.5.1 Seminumerical algorithms for linear differential equations

We promote linear differential equations as a data structure to represent and compute with functions
(see §3.1). In truth, this data structure represents functions up to finitely many constants. It determines a
global behavior but misses the pointwise aspect. Seminumerical methods combine both. They are an
important tool for experimental mathematics because they can give strong indications about the nature
of a function in very general situations (see §3.3.1).

Factorization. Alexandre Goyer and Raphaël Pagès started a PhD thesis on the factorization of differen-
tial operators. It is a fundamental operation for solving linear differential equations, or, at least, elucidate
the nature of the solutions. Goyer considers seminumerical methods. They rely on numerical evaluations
of the solutions of the differential operators to guess numerically a factorization. High precision makes it
possible to reconstruct the factors exactly, and a simple multiplication certifies the computation. Pagès
considers a discrete analogue of numerical evaluation: reduction modulo a prime number.

Effective analytic continuation. The main tool for computing high-precision evaluations of functions
or integrals is effective analytic continuation of solutions of linear differential equations. It is a form of
numerical ODE solver, specialized for linear equations and able to carry out high precision all along the
continuation path.

Numerical ODE solvers are a very classical topic in numerical analysis [69], with popular methods,
like Runge–Kutta or multistep methods. A much less known family of symbolic-numeric algorithms,
that we could call rigorous Taylor methods, originates from works of the Chudnovskys’ in the 1980s and
1990s [75, 74] and has later been developed by van der Hoeven [108] and Mezzarobba [125, 126]. This
family of algorithms only handles linear ODEs with polynomial coefficients, which is precisely the nature
of ODEs arising in the context of this document. But contrary to classical methods, they provide very
strong guarantees even in difficult situations, especially rigorous error bounds and correct behavior at
singular points, all very desirable features in experimental mathematics. Furthermore, they feature a
quasi-optimal complexity with respect to precision, meaning that one can compute easily with thousands
digits of precision: computing twice as many digits takes roughly twice as much time. This contrasts with
fixed-order methods, which cannot reach such precision. For example, to compute 10,000 digits, the
classical order four Runge–Kutta method would need typically 102500 steps. This quest for precision is
important and crucial in experimental mathematics and theoretical physics [39].

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Yet, as advanced as these algorithms may well be, they struggle with the huge ODEs coming from our
applications. The reason is easily explained: most algorithms and implementations are designed for small
operators and large precision and focuses on a quasilinear complexity with respect to precision. Our
situation is quite opposite, with large ODEs and comparatively modest precision. It may be interesting to
consider quadratic-time algorithms, with respect to precision, if the complexity with respect to the size of
the ODE gets better. This is a really blocking issue that must be addressed to enable new applications. To
solve the problem, we will endeavor to provide new software that pays attention to implement algorithms
for all regimes of degrees and orders but moderate precision.

3.5.2 Period computation

Periods are numerical integrals that can be computed to high precision with symbolic-numeric integra-
tion, even though current algorithms are far from enough to tackle real applications in algebraic geometry,
beyond the case of curves. Algorithms for computing periods of curves are mature [83, 129, 127, 80, 67]
and have been used, for example, for the computation of the endomorphism ring of genus 2 curves in
the LMFDB [81]. Algorithms in higher dimension are only emerging [89, 82, 140]. Their current status
does not make them suitable for many applications. Firstly, they are limited in generality. The articles [89,
82] deal with special double coverings of P2 or P3, with a low precision, while [140] deals with smooth
projective hypersurfaces. In terms of efficiency, we are only able to treat some lucky quartic surfaces (and
some very special quintic surfaces or cubic threefolds) for which the underlying ODEs are not too big.

With current methods, we managed to compute the periods of 180 000 quartic surfaces defined by
sparse polynomials [119]. This corpus of quartic surfaces was discovered by a random walk. Actually, we
are not able to compute (in a reasonable amount of time) the periods of a given quartic surface. So we
resorted to a random walk guided by ease of computation. This hinders severely the applicability. Yet,
this shows the feasibility of transcendental continuation to obtain algebraic invariants that are currently
unreachable by any other mean.

The seminumerical algorithms that we develop open perspectives in algebraic geometry. Some integ-
rals with algebraic origin, called periods, convey some interesting algebraic invariants. High-precision
computation may unravel them where purely algebraic methods fail [119]. These algebraic invariants are
crucial to determine the fine structure of algebraic varieties. We aim at designing algorithms to compute
periods efficiently for varieties of general interest, in particular K3 surfaces, quintic surfaces, Calabi–Yau
threefolds and cubic fourfolds.

3.5.3 Scattering amplitudes in quantum field theory

In quantum field theory, Feynman integrals appear when computing scattering amplitudes with perturb-
ative methods. In practice, computing Feynman integrals is the most effective way to obtain predictions
from a quantum field theory. Precise prediction requires higher-order perturbative terms leading to more
complex integrals and daunting computational challenges. For example, [38] reports on the methods
used, the difficulties encountered and the limitations met when computing precision calculation for
teraelectronvolt collisions in the Large Hadron Collider (LHC).

As far as mathematics is concerned, Feynman integrals are periods. Although this makes the evalu-
ation of Feynman integrals look like just a special case of symbolic-numeric integration, it would be naive
to pretend that our methods apply without effort: it is clear that the computations are so challenging that
only specialized methods may succeed. Current methods include sector decomposition [145] (where the
integration domain is decomposed in smaller pieces on which traditional numerical integral algorithms
perform well) and the use of differential equations [105] in a similar fashion to what we propose here,
namely the symbolic computation of integrals with a parameter combined with numerical ODE solving.
In the longer term, we expect that an efficient toolbox to deal with holonomic ideals would improve
computations with Feynman integrals. It is however too early to say.

In the short term, the experimental mathematics toolbox that we want to develop may be useful to
understand the geometry underlying some Feynman integrals. The typical outcome is simple analytic
formulas [53, 52] allowing for fast and precise computations. In this context, identifying key algebraic
invariants before engaging further mathematical thinking is crucial. For example, a key fact in the analysis
of a three-loop graph by [52] is the generic member of some family of K3 surfaces having Picard rank 19.

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For other graphs appear cubic fourfolds which we cannot investigate numerically at the moment. An
expected outcome of the previously exposed objectives is the computation of the periods of such varieties.
This is a first step towards a more systematic development of this interface with high-energy physics.

3.5.4 Software

Solid software foundations for effective analytic continuation (see §3.5.1) will be important for the other
tasks in this section. We use currently the part of the package ore_algebra developed by Marc Mezzarobba,
but it is a bottleneck for several algorithms. The plan for the software development (improvement of
ore_algebra, or whole new package) is not fixed yet: it depends on the nature of the algorithmic ideas that
will emerge.

4 Application domains

As already expressed in §2.3, our natural application domains are:

• Combinatorics,

• Probability theory,

• Number theory,

• Algebraic geometry,

• Statistical physics,

• Quantum mechanics.

5 Highlights of the year

5.1 A whole French year 2023 dedicated to computer algebra

The year 2023 has seen a lot of international meetings in France, in the framework of the cycle of research
schools and workshops Recent Trends in Computer Algebra 2023: 1 week in Luminy, 3 weeks in Lyon,
6 weeks in Paris, for the formal program, plus a 3-month presence at Institut Henri Poincaré while
international visitors where present.

This has led to a huge involvement of the team, both because some of the members have been part of
the organizers (Alin Bostan for the whole event and two weeks; Pierre Lairez for four weeks), and because
most of the team members have attended at least half of the 10 weeks of events.

The team has contributed almost 25 ke to the funding the events, thanks to the ERC project “10000
DIGITS” and to the ANR project De rerum natura. This has induced an increased activity of our assistant,
Bahar Carabetta.

6 New results

Participants:

Alin Bostan, Hadrien Brochet, Frédéric Chyzak, Guy Fayolle,
Pierre Lairez, Rafael Mohr, Hadrien Notarantonio, Raphaël Pagès,
Eric Pichon-Pharabod, Catherine St-Pierre, Sergey Yurkevich.

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6.1 Effective algebraicity for solutions of systems of functional equations with one

catalytic variable

In [18], Notarantonio and Yurkevich studied systems of n ≥ 1 discrete differential equations of order k ≥
1 in one catalytic variable and provided a constructive and elementary proof of algebraicity of their
solutions. This yielded effective bounds and a systematic method for computing the minimal polynomials.
Their approach is a generalization of the pioneering work by Bousquet-Mélou and Jehanne (2006).

6.2 Fast algorithms for discrete differential equations

Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series
F (t , u) in t with polynomial coefficients in a “catalytic” variable u and the specializations, say at u = 1,
of F (t , u) and of some of its partial derivatives in u. DDEs occur frequently in combinatorics, especially in
map enumeration. If a DDE is of fixed-point type then its solution F (t , u) is unique, and a general result
by Popescu (1986) implies that F (t , u) is an algebraic power series. Constructive proofs of algebraicity
for solutions of fixed-point type DDEs were proposed by Bousquet-Mélou and Jehanne (2006). In
2022, the team had initiated a systematic algorithmic study of such DDEs of order 1 in a collaboration
with Sorbonne University. In [15], Bostan and Notarantonio, together with Safey El Din (Sorbonne U.),
generalized this study to DDEs of arbitrary order. First, they proposed nontrivial extensions of algorithms
based on polynomial elimination and on the guess-and- prove paradigm. Second, they designed two
brand-new algorithms that exploit the special structure of the underlying polynomial systems. Last,
but not least, they reported on implementations that were able to solve highly challenging DDEs with a
combinatorial origin.

6.3 Systems of discrete differential equations, constructive algebraicity of the solu-

tions

In [32], Notarantonio and Yurkevich studied systems of n ≥ 1, not necessarily linear, discrete differential
equations (DDEs) of order k ≥ 1 with one catalytic variable. They provided a constructive and elementary
proof of algebraicity of the solutions of such equations. This part of the present article can be seen as
a generalization of the pioneering work by Bousquet-Mélou and Jehanne (2006) who settled down the
case n = 1. Moreover, Notarantonio and Yurkevich obtained effective bounds for the algebraicity degrees
of the solutions and provided an algorithm for computing annihilating polynomials of the algebraic series.
Finally, they carried out a first analysis in the direction of effectivity for solving systems of DDEs in view
of practical applications.

6.4 Reflected Brownian motion in a non convex cone

Guy Fayolle, in collaboration with S. Franceschi (Télécom SudParis, Institut Polytechnique de Paris) and
K. Raschel (CNRS, Université d’Angers, LAREMA), studies the stationary reflected Brownian motion in a
non-convex wedge, which, compared to its convex analogue model, has been much rarely analyzed in
the probabilistic literature. Two approaches are proposed for the three-quarter plane.

1. In [94], it was proved that the stationary distribution can be found by solving a two-dimensional
vector boundary value problem (BVP) on a single curve (a hyperbola) for the associated Laplace
transform. The reduction to this kind of vector BVP seems to be quite new in the literature. As
a matter of comparison, a single boundary condition is sufficient in the convex case. When the
parameters of the model (drift, reflection angles and covariance matrix) are symmetric with respect
to the bisector line of the cone, the model is reducible to a standard reflected Brownian motion in a
convex cone. The authors additionally constructed a one-parameter family of distributions, which
surprisingly provides for any wedge (convex or not) a particular example of stationary distribution
of a reflected Brownian motion.

2. The main result in [9] is to show that the stationary distribution can be obtained by solving a
boundary value problem of the same kind as the one encountered in the quarter plane, up to

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various dualities and symmetries. The idea is to start from Fourier (and not Laplace) transforms,
allowing to get a functional equation for a single function of two complex variables.

6.5 A Markovian analysis of IEEE 802.11 broadcast transmission networks with

buffering and back-off stages

Following up their previous analysis [96], G. Fayolle and P. Mühlethaler analyzed in [29] the so-called
back-off technique of the IEEE 802.11 protocol in broadcast mode with waiting queues. In contrast to
existing models, packets arriving when a station (or node) is in back-off state are not discarded, but are
stored in a buffer of infinite capacity. As in previous studies, the key point of their analysis hinges on
the assumption that the time on the channel is viewed as a random succession of transmission slots,
whose duration corresponds to the length of a packet, and mini-slots during which the back-off of the
station is decremented. These events occur independently, with given probabilities. The state of a node
is represented by a three-dimensional Markov chain in discrete-time, formed by the back-off counter,
the number of packets at the station, and the back-off stage. The stationary behaviour can be explicitly
solved. In particular, Fayolle and Mühlethaler obtained stability (ergodicity) conditions and interpreted
them in terms of maximum throughput.

6.6 On the representability of sequences as constant terms

A constant term sequence is a sequence of rational numbers whose n-th term is the constant term of
P n(x)Q(x), where P (x) and Q(x) are multivariate Laurent polynomials. While the generating functions
of such sequences are invariably diagonals of multivariate rational functions, and hence special period
functions, it is a famous open question, raised by Don Zagier, to classify those diagonals which are
constant terms. In [4], Alin Bostan and Sergey Yurkevich, in collaboration with Armin Straub (University
of South Alabama, USA), we provided such a classification in the case of sequences satisfying linear
recurrences with constant coefficients. They further considered the case of hypergeometric sequences
and, for a simple illustrative family of hypergeometric sequences, classified those that are constant terms.

6.7 An algorithmic approach to Rupert’s problem

A polyhedron P ⊂ R3 has Rupert’s property if a hole can be cut into it, such that a copy of P can pass
through this hole. There are several works investigating this property for some specific polyhedra: for
example, it is known that all 5 Platonic and 9 out of the 13 Archimedean solids admit Rupert’s property.
A commonly believed conjecture states that every convex polyhedron is Rupert. In the paper [13] by
Sergey Yurkevich in collaboration with Jakob Steininger (U. Vienna, Austria) it is proved that Rupert’s
problem is algorithmically decidable for polyhedra with algebraic coordinates. The authors also designed
a probabilistic algorithm which can efficiently prove that a given polyhedron is Rupert. Using this
algorithm they not only confirmed this property for the known Platonic and Archimedean solids, but also
proved it for one of the remaining Archimedean polyhedra and many others. Moreover, they significantly
improved on almost all known Nieuwland numbers and conjectured, based on statistical evidence, that
the Rhombicosidodecahedron is in fact not Rupert.

6.8 Algebraicity of hypergeometric functions with arbitrary parameters

In [30], Sergey Yurkevich, together with Florian Fürnsinn (U. Vienna), provide a complete classification
of the algebraicity of (generalized) hypergeometric functions with no restriction on the set of their
parameters. Their characterization relies on the interlacing criteria of Christol (1987) and Beukers-
Heckman (1989) for globally bounded and algebraic hypergeometric functions, however in a more
general setting which allows arbitrary complex parameters with possibly integral differences. They also
showcase the adapted criterion on a variety of different examples.

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6.9 Fast computation of the N -th term of a q-holonomic sequence and applications

(cid:112)

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial N ! in
arithmetic complexity quasi-linear in
N . In 1988, the Chudnovsky brothers generalized Strassen’s
algorithm to the computation of the N -th term of any holonomic sequence in essentially the same
arithmetic complexity. In [5], Alin Bostan and Sergey Yurkevich designed q-analogues of these algorithms.
They first extend Strassen’s algorithm to the computation of the q-factorial of N , then Chudnovskys’
algorithm to the computation of the N -th term of any q-holonomic sequence. Both algorithms work
in arithmetic complexity quasi-linear in
N ; surprisingly, they are simpler than their analogues in the
holonomic case. They provide a detailed cost analysis, in both arithmetic and bit complexity models.
Moreover, they describe various algorithmic consequences, including the acceleration of polynomial and
rational solving of linear q-differential equations, and the fast evaluation of large classes of polynomials,
including a family recently considered by Nogneng and Schost.

(cid:112)

6.10 A short proof of a non-vanishing result by Conca, Krattenthaler and Watanabe

In their 2009 paper Regular sequences of symmetric polynomials, Aldo Conca, Christian Krattenthaler and
Junzo Watanabe needed to prove, as an intermediate result, the fact that all terms of a family of binomial
sums indexed by an integer h are non-zero, except for h = 3. The proof in their paper (Appendix, pp.
190–199) performs a long and quite intricate 3-adic analysis. In [2], Alin Bostan proposes a shorter and
elementary proof.

6.11 Persistence probabilities and Mallows-Riordan polynomials

Mallows-Riordan polynomials, sometimes also called inversion polynomials, form a family of polynomials
with integer coefficients appearing in many counting problems in enumerative combinatorics. They are
also connected with the cumulant generating function of the classical log-normal distribution in probab-
ility theory. In [1] Alin Bostan, together with his probabilist co-authors Gerold Alsmeyer (U. Münster),
Kilian Raschel (CNRS, U. Angers) and Thomas Simon (U. Lille), provide a probabilistic interpretation
of the Mallows-Riordan polynomials that is not only quite different from the classical connection with
the log-normal distribution, but in fact also rather unexpected. More precisely, they establish exact for-
mulae in terms of Mallows-Riordan polynomials for the persistence probabilities of a class of order-one
autoregressive processes with symmetric uniform innovations. These exact formulae then lead to precise
asymptotics of the corresponding persistence probabilities. The connection of the Mallows-Riordan
polynomials with the volumes of certain polytopes is also discussed. Two further results provide general
factorizations of AR(1) models with continuous symmetric innovations, one for negative and one for
positive drift. The second factorization extends a classical universal formula of Sparre Andersen for
symmetric random walks.

6.12 Continued fractions, orthogonal polynomials and Dirichlet series

Using an experimental mathematics approach, Alin Bostan together with Frédéric Chapoton (CNRS, IRMA
Strasbourg) obtained in [23] new relations between the Dirichlet series for certain periodic coefficients and
the moments of certain families of orthogonal polynomials. In addition to the classical hypergeometric
orthogonal polynomials, of Racah type and continuous dual Hahn, a new similar family of orthogonal
polynomials was discovered.

6.13 Minimization of differential equations and algebraic values of E -functions

A power series being given as the solution of a linear differential equation with appropriate initial
conditions, minimization consists in finding a non-trivial linear differential equation of minimal order
having this power series as a solution. This problem exists in both homogeneous and inhomogeneous
variants; it is distinct from, but related to, the classical problem of factorization of differential operators.
Recently, minimization has found applications in Transcendental Number Theory, more specifically in
the computation of non-zero algebraic points where Siegel’s E -functions take algebraic values. In [3]

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Alin Bostan together with Bruno Salvy (Inria, ENS Lyon) and Tanguy Rivoal (CNRS, IF Grenoble) present
algorithms for these questions and discuss implementation and experiments.

6.14 A sharper multivariate Christol’s theorem with applications to diagonals and

Hadamard products

In [21], Alin Bostan, together with Boris Adamczewski (ICJ, Lyon) and Xavier Caruso (IMB, Bordeaux),
provide a new proof of the multivariate version of Christol’s theorem about algebraic power series with
coefficients in finite fields, as well as of its extension to perfect ground fields of positive characteristic
obtained independently by Denef and Lipshitz, Sharif and Woodcok, and Harase. Their new proof is
elementary, effective, and allows for much sharper estimates. They discuss various applications of such
estimates, in particular to a problem raised by Deligne concerning the algebraicity degree of reductions
modulo p of diagonals of multivariate algebraic power series with integer coefficients.

6.15 Algebraic solutions of linear differential equations: an arithmetic approach

Given a linear differential equation with coefficients in Q(x), an important question is to know whether its
full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic.
These questions are treated in [22], Alin Bostan, together with Xavier Caruso (IMB, Bordeaux) and Julien
Roques (ICJ, Lyon). After presenting motivating examples coming from various branches of mathematics,
they advertise in an elementary way a beautiful local-global arithmetic approach to these questions,
initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still
unsolved Grothendieck-Katz p-curvature conjecture.

6.16 Refined product formulas for Tamari intervals

In [24], Alin Bostan and Frédéric Chyzak, together with Vincent Pilaud (CNRS & LIX, Palaiseau) provided
short product formulas for the f -vectors of the canonical complexes of the Tamari lattices and of the
cellular diagonals of the associahedra.

6.17 Beating binary powering for polynomial matrices

The N th power of a polynomial matrix of fixed size and degree can be computed by binary powering as
fast as multiplying two polynomials of linear degree in N . When Fast Fourier Transform (FFT) is available,
the resulting complexity is softly linear in N , i.e. linear in N with extra logarithmic factors. In [14], Alin
Bostan and Sergey Yurkevich, together with Vincent Neiger (Sorbonne U.), show that it is possible to beat
binary powering, by an algorithm whose complexity is purely linear in N , even in absence of FFT. The key
result making this improvement possible is that the entries of the N th power of a polynomial matrix satisfy
linear differential equations with polynomial coefficients whose orders and degrees are independent
of N . Similar algorithms are proposed for two related problems: computing the N th term of a C-finite
sequence of polynomials, and modular exponentiation to the power N for bivariate polynomials.

6.18 On the q-analogue of Pólya’s theorem

In [6], Alin Bostan and Sergey Yurkevich answer a question posed by Michael Aissen in 1979 about the
q-analogue of a classical theorem of George Pólya (1922) on the algebraicity of (generalized) diagonals of
bivariate rational power series. In particular, they prove that the answer to Aissen’s question, in which he
considers q as a variable, is negative in general. Moreover, they show that when q is a complex number,
the answer is positive if and only if q is a root of unity.

6.19 Positivity certificates for linear recurrences

In [17], Alaa Ibrahim, together with Bruno Salvy (ARIC team), consider linear recurrences with polynomial
coefficients of Poincaré type and with a unique simple dominant eigenvalue. They give an algorithm
that proves or disproves positivity of solutions provided the initial conditions satisfy a precisely defined

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genericity condition. For positive sequences, the algorithm produces a certificate of positivity that is a
data-structure for a proof by induction. This induction works by showing that an explicitly computed
cone is contracted by the iteration of the recurrence.

6.20 A signature-based algorithm for computing the nondegenerate locus of a poly-

nomial system

Polynomial system solving arises in many application areas to model non-linear geometric properties. In
such settings, polynomial systems may come with degeneration which the end-user wants to exclude
from the solution set. The nondegenerate locus of a polynomial system is the set of points where the
codimension of the solution set matches the number of equations. Computing the nondegenerate locus
is classically done through ideal-theoretic operations in commutative algebra such as saturation ideals
or equidimensional decompositions to extract the component of maximal codimension. By exploiting
the algebraic features of signature-based Gröbner basis algorithms the authors design an algorithm
which computes a Gröbner basis of the equations describing the closure of the nondegenerate locus of a
polynomial system, without computing first a Gröbner basis for the whole polynomial system.

This is a work of Pierre Lairez, together with Christian Eder, Rafael Mohr nad Mohab Safey El Din [8].

6.21 Algorithms for minimal Picard-Fuchs operators of Feynman integrals

In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function
in projective space. By using an algorithm that extends the Griffiths–Dwork reduction for the case of
projective hypersurfaces with singularities, the authors derive Fuchsian linear differential equations,
the Picard–Fuchs equations, with respect to kinematic parameters for a large class of massive multi-
loop Feynman integrals. With this approach the authors obtain the differential operator for Feynman
integrals to high multiplicities and high loop orders. Using recent factorisation algorithms the authors
give the minimal order differential operator in most of the cases studied in this paper. Amongst our
results are that the order of Picard–Fuchs operator for the generic massive two-point n − 1-loop sunset
integral in two-dimensions is 2n − (cid:161) n+1
(cid:162) supporting the conjecture that the sunset Feynman integrals
(cid:165) n+1
(cid:166)
2
are relative periods of Calabi–Yau of dimensions n − 2. The Authors have checked this explicitly till
six loops. As well, the authors obtain a particular Picard–Fuchs operator of order 11 for the massive
five-point tardigrade non-planar two-loop integral in four dimensions for generic mass and kinematic
configurations, suggesting that it arises from K 3 surface with Picard number 11. The Authors determine
as well Picard–Fuchs operators of two-loop graphs with various multiplicities in four dimensions, finding
Fuchsian differential operators with either Liouvillian or elliptic solutions.

This is a work of Pierre Lairez, together with Pierre Vanhove, published in 2023 [12].

6.22 Factoring differential operators on algebraic curves in positive characteristic

Factorisation of linear differential operators and systems in positive characteristic have been studied
before by van der Put in [138] and Thomas Cluzeau in [78] where both made use of the p-curvature to
find a first factorisation of differential operators and systems respectively. Unfortunately the p-curvature
alone is not enough to completely factor operators as a product of irreducible differential operators when
its characteristic polynomial is not square free.
In particular to this point no algorithm where yet known to factor central operators in polynomial time.

In [132], Raphaël Pagès presented a refinement of this method to factor differential operators on
algebraic curves of positive characteristic relying on tools of algebraic geometry such as Riemann-Roch
spaces and Picard 0 group which should solve this issue. Additionally this work should allow for a better
control of the size of the output of the factorisation.

R. Pagès will defend his PhD in Spring 2024 and is preparing a corresponding submission.

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6.23 Reduction-based creative telescoping for definite summation of D-finite func-

tions

Creative telescoping is an algorithmic method introduced by Zeilberger in the 1990s to compute paramet-
rized definite sums. It proceeds by synthesizing special summands, called certificates, which have the
specific property to telescope; correspondingly, this determines a recurrence equation, called telescoper,
satisfied by the definite sum. Hadrien Brochet and Bruno Salvy (ARIC team) described a creative telescop-
ing algorithm that computes telescopers for definite sums of D-finite sequences as well as the associated
certificates in a compact form. Their algorithm relies on a discrete analogue of the generalized Hermite
reduction, or equivalently, on a generalization of the Abramov–Petkovšek reduction. They provide a Maple
implementation with good timings on a variety of examples. An article was submitted this year [26].

6.24 p-adic algorithm for bivariate Gröbner bases

Catherine St-Pierre and Éric Schost (U. Waterloo) presented a p-adic algorithm to recover the lexico-
graphic Gröbner basis G of an ideal in Q[x, y] with a generating set in Z[x, y]. The algorithm shows a
complexity that is less than cubic in terms of the dimension of Q[x, y]/〈G 〉 and is softly linear in the height
of its coefficients. Schost and St-Pierre observed that previous results of Lazard’s that use Hermite normal
forms to compute Gröbner bases for ideals with two generators can be generalised to a set of any finite
number of generators. They used the result to obtain a bound on the height of the coefficients of G , and
to control the probability of choosing a good prime p to build the p-adic expansion of G .

A long version of this manuscript [19] has been submitted.

6.25 Algorithms in intersection theory in the plane

In her PhD thesis, Catherine St-Pierre presented an algorithm to find the local structure of intersections
of plane curves. More precisely, she addressed the question of describing the scheme of the quotient ring
of a bivariate zero-dimensional ideal I ⊆ K[x, y], i.e. finding the points (maximal ideals of K[x, y]/I ) and
describing the regular functions on those points. A natural way to address this problem is via Gröbner
bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of
rings of regular functions can be studied with those bases through the division algorithm and localisation.
Let I ⊆ K[x, y] be an ideal generated by F , a subset of A[x, y] with A (cid:44)→ K and K a field. St-Pierre and Éric
Schost (U. Waterloo) presented an algorithm that features a quadratic convergence to find a Gröbner
basis of I or its primary component at the origin. The thesis presents an m-adic Newton iteration to lift
the lexicographic Gröbner basis of any finite intersection of zero-dimensional primary components of I
if m ⊆ A is a good maximal ideal. It relies on a structural result about the syzygies in such a basis due to
Conca and Valla [79], from which arises an explicit map between ideals in a stratum (or Gröbner cell) and
points in the associated moduli space. The thesis also qualified what makes a maximal ideal m suitable for
her filtration. When the field K is large enough, endowed with an Archimedean or ultrametric valuation,
and admits a fraction reconstruction algorithm, this result is used to give a complete m-adic algorithm to
recover G , the Gröbner basis of I . Inspired by previous results of Lazard’s that use Hermite normal forms
to compute Gröbner bases for ideals with two generators can be generalised to a set of n generators. This
result is then used to obtain a bound on the height of the coefficients of G and to control the probability
of choosing a good maximal ideal m ⊆ A to build the m-adic expansion of G . Inspired by Pardue [133],
a constructive proof is given to characterise a Zariski open set of GL2(K) (with action on K[x, y]) that
changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional I becomes
Borel-fixed when |K| is sufficiently large. This sharpened her analysis to obtain, when A = Z or A = k[t ], a
complexity less than cubic in terms of the dimension of Q[x, y]/〈G 〉 and softly linear in the height of the
coefficients of G . The resulting method is also adapted to extract 〈x, y〉-primary component of I . She also
discussed the transition towards other primary components via linear mappings, called untangling and
tangling, introduced by van der Hoeven and Lecerf [84]. The two maps form one isomorphism to find
points with an isomorphic local structure and, at the origin, bind them. Together with Hyun, Melczer
and Schost (U. Waterloo), she gave a slightly faster tangling algorithm and discussed new applications of
these techniques. They showed how to extend these ideas to bivariate settings and gave a bound on the
arithmetic complexity for certain algebras.

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The thesis was realised at the University of Waterloo, and the redaction was completed during St-

Pierre’s time with MATHEXP. Catherine St-Pierre successfully defended in April 2023.

6.26 Motivic geometry of 2-loop Feynman integrals

The Tardigrade graph is a two-loop Feynman graph. It describes a family of K3 surfaces, obtained as
quartic hypersurfaces in P3 with six A1 singularities generically. By means of a Lefschetz fibration of the
hypersurface, Eric Pichon-Pharabod, together with Charles Doran, Andrew Harder and Pierre Vanhove
provided [28] a description of the homology of the K3 surface that is explicit enough to perform numerical
integration on. In particular, this allows computing numerical approximations of the periods of the K3
surface with rigorous bounds, with sufficient precision to recover some algebraic invariants of the generic
family. The authors recovered the generic Picard rank of this family of K3 surfaces. Furthermore, they
obtained an explicit embedding of the Picard lattice in the full homology lattice, where the components
of the singular fibres are identified.

6.27 Estimating major merger rates and spin parameters ab initio via the clustering

of critical events

At large scales, the mass configuration of space takes a web-like structure consisting of nodes, connected
by filaments, themselves connected by walls, separated by voids. This structure and its evolution with time
have an impact on the zoology of galaxies one may find at a given point. In [27], Eric Pichon-Pharabod,
together with Corentin Cadiou, Christophe Pichon and Dmitri Pogosyan provided a model to predict this
evolution ab initio. They applied it to recover the probability distribution function of satellite-merger
separation, the distribution of mergers with respect to peak rarity, and they analysed the typical spin
brought by mergers.

6.28 Effective homology and periods of complex projective hypersurfaces

In [31], Pierre Lairez, Eric Pichon-Pharabod, together with Pierre Vanhove (IPhT) provided an algorithm to
compute an effective description of the homology of complex projective hypersurfaces relying on Picard-
Lefschetz theory. Next, they used this description to compute high-precision numerical approximations
of the periods of the hypersurface. This is an improvement over existing algorithms as this new method
allows for the computation of periods of smooth quartic surfaces in an hour on a laptop, which could
not be attained with previous methods. The general theory presented in this paper can be generalised to
varieties other than just hypersurfaces, such as elliptic fibrations as showcased on an example coming
from Feynman graphs. The algorithm comes with a SageMath implementation.

6.29 A direttissimo algorithm for equidimensional decomposition

In [16], Pierre Lairez and Rafael Mohr, together with Christian Eder (U. Kaiserslautern), and Mohab
Safey El Din (Sorbonne U.) described a recursive algorithm that decomposes an algebraic set into
locally closed equidimensional sets, i.e. sets which each have irreducible components of the same
dimension. At the core of this algorithm, they combined ideas from the theory of triangular sets, a.k.a.
regular chains, with Gröbner bases to encode and work with locally closed algebraic sets. Equipped
with this, their algorithm avoids projections of the algebraic sets that are decomposed and certain
genericity assumptions frequently made when decomposing polynomial systems, such as assumptions
about Noether position. This makes it produce fine decompositions on more structured systems where
ensuring genericity assumptions often destroys the structure of the system at hand. Practical experiments
demonstrate its efficiency compared to state-of-the-art implementations.

6.30 Integer sequences, differential operators and algebraic power series

Sergey Yurkevich successfully completed his PhD studies in July 2023. The defense of his thesis [20] took
place in Vienna and the committee consisted of Andreas Cap (president), Gilles Villard, Wadim Zudilin
(reviewers), Alin Bostan, Stéphane Fischler, Herwig Hauser (examiners).

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Inria Annual Report 2023

7 Partnerships and cooperations

7.1 International initiatives

7.1.1 Participation in other International Programs

PRCI “EAGLES”

Participants:

Alin Bostan (PI), Frédéric Chyzak.

Title: Efficient Algorithms for Guessing, inequaLitiEs and Summation

Partner Institution(s):

• JKU (Linz), Austria

• RICAM (Linz), Austria

• Inria (Saclay), France

• Sorbonne Univ. (Paris), France

Date/Duration: March 2023 – March 2027

Additionnal info/keywords: This is a PRCI ANR/FWF project between two computer algebra teams in
France and two computer algebra teams in Austria. The Austrian PI is Manuel Kauers from JKU
Linz. The goal is to work together on four axes: structured and multivariate guessing, inequalities
and D-finiteness, creative telescoping and applications in combinatorics, number theory and
theoretical physics. The obtained funding is of 770,000 euros in total, a major part of which will be
used to fund 4 PhD theses.

PhD project of Yurkevich

Title: Integer sequences, algebraic series and differential operators.

Partner Institution(s):

• University of Vienna, Austria.

Date/Duration: September 2020 – August 2023

Additionnal info/keywords: The PhD thesis project of Sergey Yurkevich is a cotutelle with the University
of Vienna (Austria). The supervisors are Alin Bostan on the French side and Herwig Hauser on the
Austrian side. The investigation topic covers on the one hand integer sequences naturally arising in
various scientific disciplines such as number theory, combinatorics and physics, and on the other
hand solutions to special kinds of differential equations.

7.2 European initiatives

7.2.1 Horizon Europe

• ERC Starting Grant “10000 DIGITS”. This project led by Pierre Lairez spans for five years starting
from April 2022. It funds three PhD theses and three 2-year post-doctoral positions. Its goal is to
develop algorithms and software to compute with high precision integrals with a geometric origin,
especially periods of algebraic varieties, and to tackle applications in diophantine approximation,
quantum field theory, and optimization.

• Postdoctoral fellowship from MathInGreaterParis. Claudia Fevola obtained a two-year postdoctoral
fellowship hosted by MATHEXP and funded by the MathInGreaterParis Fellowship Programme,
cofunded by Marie Sklodowska-Curie Actions H2020-MSCA-COFUND-2020.

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7.3 National initiatives

• De rerum natura. This project, set up by the team, was accepted this year and will be funded
until 2024. It gathers over 20 experts from four fields: computer algebra; the Galois theories
of linear functional equations; number theory; combinatorics and probability. Our goal is to
obtain classification algorithms for number theory and combinatorics, particularly so for deciding
irrationality and transcendence. (Permanent members with pm listed: Bostan, Chyzak (PI), Lairez.)

• ∂ifference. This project, led by Olivier Bournez (Lix), started in November 2020. Its objective
is to consider a novel approach in between the two worlds: discrete-oriented computations on
the one side and differential equations on the other side. We aims at providing new insights
on classical complexity theory, computability and logic through this prism and at introducing
new perspectives in algorithmic methods for differential equations solving and computer science
applications. (Permanent members with pm listed: Bostan, Chyzak.)

7.4 Regional initiatives

• Alin Bostan is co-leader of the Amadeus (Campus France) bilateral project “Integer Sequences
arising in Number Theory, Combinatorics and Physics” between France and Austria. The Austrian
co-leader is Herwig Hauser (U. Vienna, Austria).

8 Dissemination

Participants:

Alin Bostan, Hadrien Brochet, Frédéric Chyzak, Guy Fayolle, Alex-
andre Guillemot, Alaa Ibrahim, Pierre Lairez, Hadrien Notarantonio,
Raphaël Pagès, Eric Pichon-Pharabod, Sergey Yurkevich.

8.1 Promoting scientific activities

8.1.1 Scientific events: organisation

General chair, scientific chair

• Alin Bostan is part of the Scientific advisory board of the conference series Effective Methods in

Algebraic Geometry (MEGA).

• Pierre Lairez is part of the executive committee of the 2024 edition of the conference series Effective

Methods in Algebraic Geometry (MEGA).

• Since 2020, for a period of 5 years, Alin Bostan is member of the steering committee of the Journées
Nationales de Calcul Formel (JNCF), the annual meeting of the French computer algebra com-
munity.

• Alin Bostan is part of the scientific committee of the conference Combinatorics and Arithmetic for

Physics: Tenth Anniversary Edition held at IHES from Nov. 15–17, 2023.

• Alin Bostan is part of the scientific committee of the GDR EFI (“Functional Equations and Inter-
actions”) dependent on the mathematical institute (INSMI) of CNRS. The goal of this GDR is to
bring together various research communities in France working on functional equations in fields of
computer science and mathematics.

Member of the organizing committees

• Alin Bostan co-organized, with Javier Fresán and Tanguy Rivoal, the workshop E-functions, G-

functions and Periods at the Henri Poincaré Institute, 9-11 January 2023.

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Inria Annual Report 2023

• Alin Bostan co-organized, with Omid Amini, Kevin Destagnol and Pierre-Guy Plamondon, the

online summer school Mathematical Summer in Paris, 26-30 June 2023.

• Alin Bostan co-organized, with Boris Adamczewski, Bruno Salvy and Wadim Zudilin, the workshop

Effective Aspects in Diophantine Approximation, 27-31 March 2023.

• Alin Bostan organized the RTCA Special Week, Nov. 27 - Dec. 1, 2023.

• Alin Bostan and Pierre Lairez co-organized, with Jérémie Bouttier, Thomas Cluzeau, Lucia Di Vizio,
Christian Krattenthaler and Jean-Marie Maillard, the workshop Computer Algebra for Functional
Equations in Combinatorics and Physics, 4-8 December 2023.

• Alin Bostan co-organizes, with Lucia Di Vizio, the Séminaire Différentiel between U. Versailles and

Inria Saclay, with a bi-annual frequency.

• Alin Bostan co-organizes, with Lucia Di Vizio and Kilian Raschel the working group Transcendance

et Combinatoire, at Institut Henri Poincaré (Paris), with a weekly frequency.

• Alin Bostan, together with Mohab Safey El Din, Bruno Salvy and Gilles Villard, organize a thematic
program, Recent Trends in Computer Algebra (RTCA), which has been held in 2023 in Paris and Lyon.
The main funders of this event have been IHP (120 ke) and Labex Milyon (60 ke). Several of the
workshops and meeting listed above have been part of RTCA.

• Frédéric Chyzak co-organized the special session D-Finite Functions and Beyond: Algorithms,
Combinatorics, and Arithmetic as part of the international conference Applications of Computer
Algebra (ACA 2023) (Warsaw, Poland, July 2023). This was collective work with Shaoshi Chen
(Chinese Academy of Sciences, China), Antonio Jiménez-Pastor (Aalborg University, Denmark),
and Manuel Kauers and Veronika Pillwein (Johannes Kepler University Linz, Austria).

8.1.2 Scientific events: selection

Reviewer

• Alin Bostan has been a reviewer for the international conference Formal Power Series and Algebraic

Combinatorics.

• Frédéric Chyzak has served as a reviewer for the International Symposium on Symbolic and Algebraic

Computation (ISSAC’23).

• Hadrien Notarantonio has been a reviewer for the international conference Formal Power Series

and Algebraic Combinatorics.

8.1.3 Journal

Member of the editorial boards

• Alin Bostan is on the editorial boards of Journal of Symbolic Computation, the Annals of Combin-
atorics, the Maple Transactions and the Bulletin of the Transilvania University of Bras, ov, Series III:
Mathematics and Computer Science.

• Frédéric Chyzak is on the editorial board of the Journal of Systems Science and Complexity (JSSC).

• Guy Fayolle is associate editor of the journal Markov Processes and Related Fields (MPRF).

• Pierre Lairez is on the editorial board of the SIAM Journal in applied algebraic geometry (SIAGA).

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Reviewer - reviewing activities

• Alin Bostan has been a reviewer for the international journals: Annals of Combinatorics; Combinat-
orial Theory; Indagationes Mathematicae; Journal of Symbolic Computation. He also served as a
reviewer for a book published by Springer in the series Algorithms and Computation in Mathemat-
ics.

• Frédéric Chyzak has been a reviewer for the journals MathematicS In Action and Journal of the
London Mathematical Society, as well as for the AMS book series Mathematical Surveys and Mono-
graphs.

• Guy Fayolle has been a reviewer for: Markov Processes and Related Fields; Physica A: Statistical

Mechanics and its Applications; Transactions of the American Mathematical Society.

• Sergey Yurkevich has been a reviewer for the journals Geometriae Dedicata and The Fibonacci

Quarterly.

• Eric Pichon-Pharabod has been a reviewer for the international journal Annals of Combinatorics.

8.1.4 Invited talks

• Alin Bostan gave a talk at the workshop “Periods” (February 2023, Lisbon, Portugal).

• Alin Bostan gave a talk at the seminar “Équations différentielles motiviques et au-delà” (April 2023,

IHP, Paris, France).

• Alin Bostan gave a talk at the conference FoCM 2023 (June 2023, Paris, France).

• Alin Bostan was invited speaker at the “Special day on commutative algebra and ordinary differen-

tial equations” (Dec. 2023, U. Vienna, Austria).

• Hadrien Brochet gave a talk at the conference Journées Nationales du Calcul Formel (March 2023,

CIRM, Luminy, France).

• Hadrien Brochet gave a talk at the conference The 28th International Conference on Applications of

Computer Algebra (July 2023, Warsaw, Poland).

• Frédéric Chyzak was invited to speak at Fundamental Algorithms and Algorithmic Complexity,
part of RTCA, where he presented ongoing work on computing first-order factors of linear Mahler
operators (Sep. 2023, Paris).

• Guy Fayolle was invited to the conference 40 Years of Reflected Brownian Motion and Related Topics,

held in Roscoff (April 2023).

• Alaa Ibrahim gave a talk at the SIAM Conference on Applied Algebraic Geometry (AG23) (July 2023,

Eindhoven, Netherlands).

• Alaa Ibrahim gave a talk at the conference Journées Nationales du Calcul Formel (March 2023,

CIRM, Luminy, France).

• Pierre Lairez gave a talk at Journées nationales du GDR IM (Paris, April 2023).

• Pierre Lairez gave a talk at the workshop Effective methods in diophantine approximation workshop

(Lyon, March 2023).

• Pierre Lairez gave a talk at the conference FoCM 2023 (June 2023, Paris, France).

• Pierre Lairez gave a talk at the Atelier E-fonctions, G-fonctions et périodes (January 2023, Paris).

• Hadrien Notarantonio gave a talk at the workshop “Periods” (February 2023, Lisbon, Portugal).

• Hadrien Notarantonio gave a talk at the conference Journées Nationales du Calcul Formel (March

2023, CIRM, Luminy, France).

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Inria Annual Report 2023

• Hadrien Notarantonio gave a talk at the conference Formal Power Series and Algebraic Combinator-

ics 2024 (July 2023, Davis, USA).

• Hadrien Notarantonio gave a talk at the conference International Symposium on Symbolic &

Algebraic Computation (July 2023, Tromsø, Norway).

• Hadrien Notarantonio gave a talk at the IHP semester Recent Trends in Computer Algebra: related

event (IHP, Paris, France).

• Hadrien Notarantonio was invited speaker at the conference Combinatorics and Arithmetic for

Physics (IHES, Bures-sur-Yvette, France).

• Hadrien Notarantonio was invited speaker at the topical day Elimination for Functional Equations,
as part of the workshop Computer Algebra for Functional Equations in Combinatorics and Physics
(IHP, Paris, France).

• Raphaël Pagès gave a talk at the conference Journées Nationales du Calcul Formel (March 2023,

CIRM, Luminy, France).

• Raphaël Pagès gave a talk at the Séminaire de Théorie des Nombres de l’Université de Caen (Decem-

ber 2023, Caen, France).

• Eric Pichon-Pharabod gave a talk at the program The Cosmic Web: Connecting Galaxies to Cosmo-
logy at High and Low Redshift (February 2023, KITP, Santa-Barbara, United States of America).

• Eric Pichon-Pharabod gave a talk at the conference Journées Nationales du Calcul Formel (March

2023, CIRM, Marseilles, France).

• Eric Pichon-Pharabod gave a talk at the SIAM Conference on Applied Algebraic Geometry (AG23)

(June 2023, Eindhoven, Netherlands).

• Eric Pichon-Pharabod was invited speaker at the Séminaire de Géométrie Algébrique de l’Université

d’Angers (September 2023, Angers, France).

• Eric Pichon-Pharabod was invited speaker at the conference Combinatorics and Arithmetic for

Physics workshop (November 2023, IHES, Bures-sur-Yvette, France).

• Eric Pichon-Pharabod was invited speaker at the Seminar series on motives and period integrals in

Quantum field theory and String theory (November 2023, Oxford, United Kingdom).

• Catherine St-Pierre gave a talk at the conference International Symposium on Symbolic & Algebraic

Computation (July 2023, Tromsø, Norway).

• Catherine St-Pierre gave a talk at the conference The 28th International Conference on Applications

of Computer Algebra (July 2023, Warsaw, Poland).

• Sergey Yurkevich gave a talk at the conference Enumerative Combinatorics (December 2022, Ober-

wolfach, Germany).

• Sergey Yurkevich gave a talk at the workshop “Periods” (February 2023, Lisbon, Portugal).

• Sergey Yurkevich gave a talk at the conference Journe‘es Nationales du Calcul Formel (March 2023,

CIRM, Luminy, France).

• Sergey Yurkevich gave a talk at the conference FoCM2023 (June 2023, Paris, France).

• Sergey Yurkevich gave a talk at the conference International Symposium on Symbolic & Algebraic

Computation (July 2023, Tromsø, Norway).

• Alexandre Goyer was invited speaker at AriC seminar (January 2023, Lyon, France).

• Alexandre Goyer was invited speaker at the séminaire d’Analyse (February 2023, Strasbourg,

France).

• Alexandre Goyer gave a remote talk at the SIAM Conference on Applied Algebraic Geometry (AG23)

(July 2023, Eindhoven, Netherlands).

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8.1.5 Scientific expertise

• Alin Bostan was part of the hiring committee for an Assistant Professor (Maitre de Conférences)

position in computer science at UGA-LJK, Grenoble U., March – April, 2023.

• Alin Bostan was part of the hiring committee for an Assistant Professor (Maitre de Conférences)

position in computer science at LIP6, Sorbonne U., April – May, 2023.

• Frédéric Chyzak has been a member of a recruitment committee for a full-time Monge assistant

professor in computer science at École Polytechnique.

8.1.6 Research administration

• Alin Bostan is one of the two members of the scientific search committee of the Inria Saclay Center.

• Frédéric Chyzak is a member of the commission of users of computer resources (CUMI) at the Inria
Saclay Center and of the commission managing the mentoring program at the Inria Saclay Center.

• Frédéric Chyzak is also a mentor in the mentoring program.

• Since 2023, Frédéric Chyzak is an elected member of the evaluation commission (CE) of Inria.

• Guy Fayolle is scientific advisor and associate researcher at the Centre for Robotics (Mines Paris

PSL).

• Guy Fayolle is a member of the working group WG 7.3: Computer System Modeling of the Interna-

tional Federation for Information Processing (IFIP).

• Hadrien Notarantonio is one of the 8 elected representatives of the doctoral school EDMH.

8.2 Teaching - Supervision - Juries

• Bachelor:

– Hadrien Brochet, Topology and multivariable calculus (MAA202), 64h, 3rd semester Bachelor

Polytechnique, France.

– Alaa Ibrahim, Cours d’analyse, 48h, L1, Université de Lyon 1, France.

– Alexandre Guillemot, Linear Algebra (MAA101), 64h, 1st semester, Bachelor Polytechnique,

France.

– Hadrien Notarantonio, Structure de données (LU2IN006), 64h, 2nd semester, Sorbonne Uni-

versité, France.

– Raphaël Pagès, Mathématiques générales and Outils Mathématiques, 90h, L1, Universite‘ de

Bordeaux, France.

– Eric Pichon-Pharabod, Topology and multivariable calculus (MAA202), 64h, 3rd semester

Bachelor Polytechnique, France.

– Eric Pichon-Pharabod, Introduction to Analysis (MAA102), 64h, 1st semester Bachelor Poly-

technique, France.

• Master:

– Alin Bostan, Algorithmes efficaces en calcul formel, 22.5h, M2, MPRI, France.

– Alin Bostan, Modern Algorithms for Symbolic Summation and Integration, 18h, M2, ENS Lyon,

France.

– Pierre Lairez, Algorithmes efficaces en calcul formel, 12h, M2, MPRI, France.

– Pierre Lairez, Competitive programming (INF473A), TD, 40h, M2, École polytechnique, France.

– Pierre Lairez, Les bases de la programmation et de l’algorithmique (INF411), TD, 40h, M1,

École polytechnique, France.

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Inria Annual Report 2023

8.2.1 Supervision

• Master interships:

– Alin Bostan co-supervised together with Bruno Salvy (Inria, ENS Lyon) the Master thesis of

Louis Gaillard on the topic “Bounds for output size of algorithms in computer algebra”.

– Alin Bostan co-supervised together with Tanguy Rivoal (Institut Fourier, Grenoble) the Master

thesis of Jordan Béraud on the topic “Factorisation d’opérateurs différentiels”.

– Pierre Lairez supervised the master thesis of Alexandre Guillemot on the topic “Certified

algebraic path tracking and braid computation”.

• PhD theses:

– Alin Bostan co-supervised together with Herwig Hauser (U. Vienna, Austria) the PhD thesis of
Sergey Yurkevich on “Integer sequences arising in number theory, combinatorics and physics”
(defended on July 6, 2023).

– Alin Bostan co-supervises together with Xavier Caruso (CNRS, IMB Bordeaux) the PhD thesis
of Raphaël Pagès on “Algorithms for factoring linear differential operators in positive charac-
teristic”.

– Alin Bostan co-supervises together with Mohab Safey El Din (Sorbonne U.) and Bruno Salvy
(Inria, ENS Lyon) the PhD thesis of Alaa Ibrahim on “Automated proofs of inequalities between
special sequences and functions”.

– Alin Bostan and Frédéric Chyzak co-supervise together with Mohab Safey El Din (Sorbonne
U.) the PhD thesis of Hadrien Notarantonio on “Geometry-driven algorithms for the efficient
solving of combinatorial functional equations”.

– Frédéric Chyzak co-supervises together with Marc Mezzarobba (CNRS, LIX) the PhD thesis of

Alexandre Goyer on “Symbolic-numeric algorithms in differential algebra”.

– Frédéric Chyzak and Pierre Lairez co-supervise the PhD thesis of Hadrien Brochet on “Al-

gorithms for D-modules”.

– Frédéric Chyzak co-supervises together with Shaoshi Chen (CAS, Beijing) the PhD thesis of

Pingchuan Ma. Pingchuan Ma was visiting France for 6 months this year.

– Pierre Lairez co-supervises together with Pierre Vanhove (CEA, IPhT) the PhD thesis of Eric
Pichon-Pharabod on “Periods in algebraic geometry: computation and application to Feyn-
man’s integrals”.

– Pierre Lairez co-supervises together with Christian Eder (TU Kaiserslautern) and Mohab Safey
El Din (Sorbonne U.) the PhD thesis of Rafael Mohr on “Equidimensional decomposition
algorithms with signature bases”.

– Pierre Lairez supervises the PhD thesis of Alexandre Guillemot on “Effective topology of

complex algebraic varieties”.

8.2.2 Juries

• Alin Bostan has served as a reviewer of the Habilitation thesis of Michael Wallner, Combinatorial

analysis of directed acyclic graphs, Young tableaux, and lattice paths, U. Vienna, Austria.

• Alin Bostan has served as a reviewer in the mid-PhD examination of Subhayan Saha, Lower bounds

and reconstruction algorithms for arithmetic circuits, ENS Lyon, Apr. 5, 2023.

• Alin Bostan was examiner in the Habilitation thesis defense jury of François Olivier, Effective formal

resolution of systems of algebraic differential equations, Ecole polytechnique, May 11, 2023.

• Alin Bostan has served as a reviewer in the mid-PhD examination of Hippolyte Signargout, Calcul
exact avec des matrices quasiséparables et des matrices polynomiales à structure de déplacement,
ENS Lyon, May 15, 2023.

Project MATHEXP

33

• Alin Bostan has served as a reviewer in the mid-PhD examination of Damien Vidal, Analyse al-

gébrique des schémas cryptographiques post-quantiques, Amiens, Sept. 15, 2023.

• Alin Bostan was examiner in the Habilitation thesis defense jury of Jérémy Berhomieu, Contribu-
tions to polynomial system solving: Recurrences and Gröbner bases, Sorbonne U., September 21,
2023.

• Alin Bostan has served as a reviewer of the PhD thesis of Hippolyte Signargout, Calcul exact avec
des matrices quasiséparables et des matrices polynomiales à structure de déplacement, ENS Lyon,
October 5, 2023. He had previously served as a reviewer in his mid-PhD examination on May 15,
2023.

• Guy Fayolle was external examiner for the PhD defense of Vasilii Goriachkin (Critical Scaling in

Particle Systems and Random Graphs), defended at Lund University (Sweden).

8.3 Scientific animation of the project-team

The team runs a monthly seminar, jointly with colleagues in Sorbonne Université, in alternation in
Palaiseau and Paris, and open to remote attendance in hybrid mode. The organizers for our team are
Pierre Lairez and Hadrien Notarantonio. Because of the huge activity induced by the semester Recent
Trends in Computer Algebra 2023, we decided to reduce the activity in our seminar during the past year.
Nevertheless, we could have 9 talks, including some by international visiting colleagues.

9 Scientific production

9.1 Publications of the year

International journals

[1] G. Alsmeyer, A. Bostan, K. Raschel and T. Simon. ‘Persistence for a class of order-one autoregressive
processes and Mallows-Riordan polynomials’. In: Advances in Applied Mathematics (2023). URL:
https://inria.hal.science/hal-03469594.

[2] A. Bostan. ‘A short proof of a non-vanishing result by Conca, Krattenthaler and Watanabe’. In:
Rendiconti del Seminario Matematico della Università di Padova (14th Jan. 2023). DOI: 10.4171
/RSMUP/113. URL: https://inria.hal.science/hal-03216337.

[3] A. Bostan, T. Rivoal and B. Salvy. ‘Minimization of differential equations and algebraic values of
E -functions’. In: Mathematics of Computation (2023). URL: https://hal.science/hal-03771
150.

[4] A. Bostan, A. Straub and S. Yurkevich. ‘On the representability of sequences as constant terms’.
In: Journal of Number Theory 253 (Dec. 2023), pp. 235–256. DOI: 10.1016/j.jnt.2023.06.015.
URL: https://hal.science/hal-03907115.

[5] A. Bostan and S. Yurkevich. ‘Fast Computation of the N -th Term of a q-Holonomic Sequence and
Applications’. In: Journal of Symbolic Computation 115 (2023), pp. 96–123. DOI: 10.1016/j.jsc
.2022.07.008. URL: https://inria.hal.science/hal-03084680.

[6] A. Bostan and S. Yurkevich. ‘On the q-analogue of Pólya’s Theorem’. In: The Electronic Journal of

Combinatorics (31st Mar. 2023). URL: https://hal.science/hal-03425997.

[7] P. Bürgisser, F. Cucker and P. Lairez. ‘Rigid continuation paths II. Structured polynomial systems’.
In: Forum of Mathematics, Pi 11 (14th Apr. 2023), e12. DOI: 10.1017/fmp.2023.7. URL: https:
//hal.science/hal-02974062.

[8] C. Eder, P. Lairez, R. Mohr and M. Safey El Din. ‘A Signature-based Algorithm for Computing the
Nondegenerate Locus of a Polynomial System’. In: Journal of Symbolic Computation 119 (Nov.
2023), pp. 1–21. DOI: 10.1016/j.jsc.2023.02.001. URL: https://hal.science/hal-03590
675.

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Inria Annual Report 2023

[9] G. Fayolle, S. Franceschi and K. Raschel. ‘Stationary brownian motion in a 3/4-plane: Reduction
to a riemann-hilbert problem via fourier transforms’. In: Indagationes Mathematicae 34.5 (Sept.
2023), 17 (874–890). DOI: 10.1016/j.indag.2022.10.008. URL: https://hal.science/hal-
03832431.

[10] P. Lairez. ‘Axioms for a theory of signature bases’. In: Journal of Symbolic Computation (2023). URL:

https://hal.science/hal-03830003.

[11] P. Lairez and E. C. Sertöz. ‘Separation of periods of quartic surfaces’. In: Algebra & Number Theory
17.10 (19th Sept. 2023), pp. 1753–1778. DOI: 10.2140/ant.2023.17.1753. URL: https://hal.s
cience/hal-03022612.

[12] P. Lairez and P. Vanhove. ‘Algorithms for minimal Picard-Fuchs operators of Feynman integrals’.
In: Letters in Mathematical Physics 113.2 (29th Mar. 2023), p. 37. DOI: 10.1007/s11005-023-016
61-3. URL: https://hal.science/hal-03784522.

[13]

J. Steininger and S. Yurkevich. ‘An algorithmic approach to Rupert’s problem’. In: Mathematics of
Computation (2023). URL: https://inria.hal.science/hal-03504867.

International peer-reviewed conferences

[14] A. Bostan, V. Neiger and S. Yurkevich. ‘Beating binary powering for polynomial matrices’. In: ISSAC
2023 - 48th International Symposium on Symbolic and Algebraic Computation. Tromso, Norway,
24th July 2023. URL: https://inria.hal.science/hal-03979664.

[15] A. Bostan, H. Notarantonio and M. Safey El Din. ‘Fast Algorithms for Discrete Differential Equa-
tions’. In: ISSAC 2023 - 48th International Symposium on Symbolic and Algebraic Computation.
Tromso, Norvège, Norway, 24th July 2023. URL: https://inria.hal.science/hal-03978089.

[16] C. Eder, P. Lairez, R. Mohr and M. Safey El Din. ‘A Direttissimo Algorithm for Equidimensional
Decomposition’. In: ISSAC ’23: Proceedings of the 2023 International Symposium on Symbolic and
Algebraic Computation. ISSAC 2023 - 48th International Symposium on Symbolic and Algebraic
Computation. Tromsø, Norway, 9th Feb. 2023. URL: https://hal.science/hal-04000248.

[17] A. Ibrahim and B. Salvy. ‘Positivity certificates for linear recurrences’. In: Proceedings SODA. SODA.
Alexandria, Virginia, United States, 2024. URL: https://inria.hal.science/hal-04271203.

[18] H. Notarantonio and S. Yurkevich. ‘Effective algebraicity for solutions of systems of functional
equations with one catalytic variable’. In: 35th international conference on Formal Power Series
and Algebraic Combinatorics (FPSAC 2023). Davis, CA, United States, 2023. URL: https://inria
.hal.science/hal-03854264.

Edition (books, proceedings, special issue of a journal)

[19] É. Schost and C. St-Pierre, eds. p-adic algorithm for bivariate Gröbner bases. ISSAC 2023: Interna-
tional Symposium on Symbolic and Algebraic Computation 2023. ACM, 24th July 2023, pp. 508–
516. DOI: 10.1145/3597066.3597086. URL: https://hal.science/hal-04360580.

Doctoral dissertations and habilitation theses

[20] S. Yurkevich. ‘Integer sequences, differential operators and algebraic power series’. Université
Paris-Saclay; Universität Wien, 6th July 2023. URL: https://theses.hal.science/tel-04209
876.

Reports & preprints

[21] B. Adamczewski, A. Bostan and X. Caruso. A sharper multivariate Christol’s theorem with applica-
tions to diagonals and Hadamard products. 5th June 2023. URL: https://hal.science/hal-04
116793.

[22] A. Bostan, X. Caruso and J. Roques. Algebraic solutions of linear differential equations: an arith-

metic approach. 11th Apr. 2023. URL: https://hal.science/hal-04065092.

Project MATHEXP

35

[23] A. Bostan and F. Chapoton. Continued fractions, orthogonal polynomials and Dirichlet series.

10th July 2023. URL: https://hal.science/hal-03832808.

[24] A. Bostan, F. Chyzak and V. Pilaud. Refined product formulas for Tamari intervals. 2023. URL:

https://inria.hal.science/hal-04102398.

[25] A. Bostan, L. Di Vizio and K. Raschel. Differential transcendence of Bell numbers and relatives: a
Galois theoretic approach. 24th Aug. 2023. URL: https://hal.science/hal-03091272.

[26] H. Brochet and B. Salvy. Reduction-Based Creative Telescoping for Definite Summation of D-Finite

Functions. 14th July 2023. URL: https://hal.science/hal-04295759.

[27] C. Cadiou, E. Pichon-Pharabod, C. Pichon and D. Pogosyan. Estimating major merger rates and
spin parameters ab initio via the clustering of critical events. 17th Oct. 2023. URL: https://hal.s
cience/hal-04246812.

[28] C. F. Doran, A. Harder, P. Vanhove and E. Pichon-Pharabod. Motivic geometry of two-loop Feynman

integrals. 21st Mar. 2023. URL: https://hal.science/hal-04038863.

[29] G. Fayolle and P. Muhlethaler. Wifi with buffering in the IEEE-802.11 protocol : a Markovian analysis
of the access scheme. 22nd Dec. 2023. URL: https://inria.hal.science/hal-04362437.

[30] F. Fürnsinn and S. Yurkevich. Algebraicity of hypergeometric functions with arbitrary parameters.

25th Aug. 2023. URL: https://inria.hal.science/hal-04205544.

[31] P. Lairez, E. Pichon-Pharabod and P. Vanhove. Effective homology and periods of complex projective

hypersurfaces. 8th June 2023. URL: https://hal.science/hal-04144264.

[32] H. Notarantonio and S. Yurkevich. Systems of Discrete Differential Equations, Constructive Algeb-

raicity of the Solutions. 19th Oct. 2023. URL: https://inria.hal.science/hal-04250469.

9.2 Cited publications

[33] S. A. Abramov, M. A. Barkatou and M. van Hoeij. ‘Apparent singularities of linear difference
equations with polynomial coefficients’. In: Applicable Algebra in Engineering, Communication
and Computing 17.2 (2006), pp. 117–133. DOI: 10.1007/s00200-005-0193-9.

[34]

S. A. Abramov and M. van Hoeij. ‘Desingularization of linear difference operators with polynomial
coefficients’. In: ISSAC ’99. Ed. by B. Salvy. Conference proceedings. ACM, 1999. DOI: 10.1145/30
9831.309953.

[35] B. Adamczewski and C. Faverjon. ‘Méthode de Mahler, transcendance et relations linéaires: aspects

effectifs’. In: Journal de Théorie des Nombres de Bordeaux 30.2 (2018), pp. 557–573.

[36] B. Adamczewski and T. Rivoal. ‘Exceptional values of E -functions at algebraic points’. In: Bulletin

of the London Mathematical Society 50.4 (2018), pp. 697–908.

[37] M. F. Adamer, A. C. L˝orincz, A.-L. Sattelberger and B. Sturmfels. ‘Algebraic analysis of rotation

data’. In: Algebraic Statistics 11.2 (28th Dec. 2020), pp. 189–211. DOI: 10/gmsgjn.

[38] S. Badger, J. Bendavid, V. Ciulli, A. Denner, R. Frederix, M. Grazzini, J. Huston, M. Schönherr,
K. Tackmann, J. Thaler et al. Les Houches 2015: Physics at TeV colliders standard model working
group report. 16th May 2016. arXiv: 1605.04692.

[39] D. Bailey and J. M. Borwein. ‘High-precision numerical integration: progress and challenges’. In:

Journal of Symbolic Computation 46.7 (2011), pp. 741–754. DOI: 10/bbf6j9.

[40] D. Bailey, P. Borwein and S. Plouffe. ‘On the rapid computation of various polylogarithmic con-
stants’. In: Mathematics of Computation 66.218 (1997), pp. 903–913. DOI: 10.1090/S0025-5718-
97-00856-9.

[41] M. Barkatou, T. Cluzeau, L. Di Vizio and J.-A. Weil. ‘Computing the Lie algebra of the differential
Galois group of a linear differential system’. In: ISSAC ’2016. Conference proceedings. ACM, 2016,
pp. 63–70.

[42] B. Beckermann and G. Labahn. ‘A uniform approach for the fast computation of matrix-type Padé
approximants’. In: SIAM Journal of Matrix Analysis and Applications 15.3 (1994), pp. 804–823. DOI:
10.1137/S0895479892230031.

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[43]

J. P. Bell and M. Coons. ‘Transcendence tests for Mahler functions’. In: Proceedings of the AMS
145.3 (2017), pp. 1061–1070.

[44] O. Bernardi and M. Bousquet-Mélou. ‘Counting colored planar maps: algebraicity results’. In:
Journal of Combinatorial Theory. Series B 101.5 (2011), pp. 315–377. DOI: 10.1016/j.jctb.2011
.02.003. URL: https://doi.org/10.1016/j.jctb.2011.02.003.

[45] O. Bernardi, M. Bousquet-Mélou and K. Raschel. ‘Counting quadrant walks via Tutte’s invariant

method’. In: Combinatorial Theory 1 (2021). DOI: 10.5070/C61055360.

[46]

[47]

J. Berthomieu, C. Eder and M. Safey El Din. ‘Msolve: a library for solving polynomial systems’. In:
ISSAC ’21. Conference proceedings. ACM, 18th July 2021, pp. 51–58. DOI: 10/gk8549.

J. Berthomieu and J.-C. Faugère. ‘A polynomial-division-based algorithm for computing linear
recurrence relations’. In: ISSAC ’18. Conference proceedings. 2018, pp. 79–86.

[48] D. Bertrand and F. Beukers. ‘Équations différentielles linéaires et majorations de multiplicités’. In:

Annales Scientifiques de l’École Normale Supérieure 18.1 (1985), pp. 181–192.

[49] F. Beukers. ‘A note on the irrationality of ζ(2) and ζ(3)’. In: Bulletin of the London Mathematical

Society 11.3 (1979), pp. 268–272.

[50] F. Beukers. ‘A refined version of the Siegel-Shidlovskii theorem’. In: Annals of Mathematics 163.1
(2006), pp. 369–379. DOI: 10.4007/annals.2006.163.369. URL: https://doi.org/10.4007
/annals.2006.163.369.

[51] F. Beukers. ‘Padé-approximations in number theory’. In: Padé Approximation and its Applications,

Amsterdam 1980. Vol. 888. Lecture Notes in Math. Springer, 1981, pp. 90–99.

[52]

S. Bloch, M. Kerr and P. Vanhove. ‘A Feynman integral via higher normal functions’. In: Compositio
Mathematica 151.12 (2015), pp. 2329–2375. DOI: 10/f74v85.

[53] S. Bloch and P. Vanhove. ‘The elliptic dilogarithm for the sunset graph’. In: Journal of Number

Theory 148 (2015), pp. 328–364. DOI: 10/gh3qps.

[54]

J. Borwein and D. Bailey. Mathematics by experiment. Plausible reasoning in the 21st century. A K
Peters, 2008, pp. xii+377.

[55] A. Bostan, F. Chyzak, M. Giusti, R. Lebreton, G. Lecerf, B. Salvy and É. Schost. Algorithmes efficaces

en calcul formel. French. 686 pages. CreateSpace, 2017.

[56] A. Bostan, C.-P. Jeannerod, C. Mouilleron and É. Schost. ‘On matrices with displacement structure:
generalized operators and faster algorithms’. In: SIAM Journal of Matrix Analysis and Applications
38.3 (2017), pp. 733–775. DOI: 10.1137/16M1062855. URL: https://doi.org/10.1137/16M10
62855.

[57] A. Bostan and M. Kauers. ‘The complete generating function for Gessel walks is algebraic’. In:
Proceedings of the AMS 138.9 (2010). With an appendix by Mark van Hoeij, pp. 3063–3078. DOI:
10.1090/S0002-9939-2010-10398-2.

[58] A. Bostan, F. Chyzak, P. Lairez and B. Salvy. ‘Generalized Hermite reduction, creative telescoping
and definite integration of D-finite functions’. In: ISSAC ’18. Conference proceedings. ACM, 2018,
pp. 95–102. DOI: 10/ddv8.

[59] A. Bostan, F. Chyzak, G. Lecerf, B. Salvy and É. Schost. ‘Differential equations for algebraic func-

tions’. In: ISSAC ’07. Conference proceedings. 2007, pp. 25–32.

[60] A. Bostan, P. Lairez and B. Salvy. ‘Creative telescoping for rational functions using the Griffiths–
Dwork method’. In: ISSAC ’13. Conference proceedings. ACM, 2013, pp. 93–100. DOI: 10/ggcmbk.

[61] A. Bostan, P. Lairez and B. Salvy. ‘Multiple binomial sums’. In: Journal of Symbolic Computation
80.part 2 (2017), pp. 351–386. DOI: 10.1016/j.jsc.2016.04.002. URL: https://doi.org/10
.1016/j.jsc.2016.04.002.

[62] A. Bostan, T. Rivoal and B. Salvy. ‘Explicit degree bounds for right factors of linear differential
operators’. In: Bulletin of the London Mathematical Society 53.1 (2020), pp. 53–62. DOI: 10.1112
/blms.12396.

Project MATHEXP

37

[63] M. Bousquet-Mélou. ‘Square lattice walks avoiding a quadrant’. In: Journal of Combinatorial

Theory. Series A 144 (2016), pp. 37–79.

[64] M. Bousquet-Mélou and A. Jehanne. ‘Polynomial equations with one catalytic variable, algebraic
series and map enumeration’. In: Journal of Combinatorial Theory. Series B 96.5 (2006), pp. 623–
672.

[65] B. Boyer, C. Eder, J.-C. Faugère, S. Lachartre and F. Martani. ‘GBLA: Gröbner basis linear algebra
package’. In: ISSAC ’16. Conference proceedings. ACM, 2016, pp. 135–142. DOI: 10/ggck6t.

[66] R. Brak and A. J. Guttmann. ‘Algebraic approximants: a new method of series analysis’. In: Journal

of Physics. A 23.24 (1990), pp. L1331–L1337. DOI: 10.1088/0305-4470/23/24/008.

[67] N. Bruin, J. Sijsling and A. Zotine. ‘Numerical computation of endomorphism rings of Jacobians’.

In: The Open Book Series 2.1 (2019), pp. 155–171. DOI: 10/ggck8d.

[68] M. Buchacher, M. Kauers and G. Pogudin. ‘Separating variables in bivariate polynomial ideals’. In:

ISSAC ’20. Conference proceedings. ACM, 2020, pp. 54–61. DOI: 10.1145/3373207.3404028.

[69]

[70]

J. C. Butcher. Numerical methods for ordinary differential equations. Third. John Wiley & Sons,
2016, pp. xxiii+513. DOI: 10/ghn6z6.

S. Chen, M. Jaroschek, M. Kauers and M. F. Singer. ‘Desingularization explains order-degree curves
for Ore operators’. In: ISSAC ’13. Conference proceedings. ACM, 2013, pp. 157–164.

[71] S. Chen and M. Kauers. ‘Order-degree curves for hypergeometric creative telescoping’. In: IS-

SAC ’12. Conference proceedings. ACM, 2012, pp. 122–129.

[72]

[73]

S. Chen and M. Kauers. ‘Trading order for degree in creative telescoping’. In: Journal of Symbolic
Computation 47.8 (2012), pp. 968–995. DOI: 10.1016/j.jsc.2012.02.002.

S. Chen, M. Kauers, Z. Li and Y. Zhang. ‘Apparent singularities of D-finite systems’. In: Journal of
Symbolic Computation 95 (2019), pp. 217–237. DOI: 10.1016/j.jsc.2019.02.009.

[74] D. V. Chudnovsky and G. V. Chudnovsky. ‘Computer algebra in the service of mathematical physics
and number theory’. In: Computers in Mathematics (Stanford, CA, 1986). Vol. 125. Lecture Notes
in Pure and Appl. Math. Dekker, 1990, pp. 109–232.

[75] D. V. Chudnovsky and G. V. Chudnovsky. ‘Computer assisted number theory with applications’.
In: Number theory (New York, 1984–1985). Ed. by D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn
and M. B. Nathanson. Vol. 1240. Lecture Notes in Mathematics. Springer, 1987, pp. 1–68. DOI:
10/dnnwtb.

[76] F. Chyzak, T. Dreyfus, P. Dumas and M. Mezzarobba. ‘Computing solutions of linear Mahler

equations’. In: Mathematics of Computation 87.314 (2018), pp. 2977–3021.

[77] F. Chyzak and P. Dumas. ‘A Gröbner-basis theory for divide-and-conquer recurrences’. In: IS-

SAC ’20. Conference proceedings. ACM, July 2020. DOI: 10.1145/3373207.3404055.

[78] T. Cluzeau. ‘Factorization of differential systems in characteristic p’. In: Proceedings of the 2003
International Symposium on Symbolic and Algebraic Computation. ACM, New York, 2003, pp. 58–
65. DOI: 10.1145/860854.860875. URL: https://doi.org/10.1145/860854.860875.

[79] A. Conca and G. Valla. ‘Canonical Hilbert-Burch matrices for ideals of k[x, y]’. In: Michigan
Mathematical Journal 57 (2008), pp. 157–172. DOI: 10.1307/mmj/1220879402. URL: https://d
oi.org/10.1307/mmj/1220879402.

[80] E. Costa, N. Mascot, J. Sijsling and J. Voight. ‘Rigorous computation of the endomorphism ring of
a Jacobian’. In: Mathematics of Computation 88.317 (2019), pp. 1303–1339. DOI: 10/ggck8g.

[81]

J. Cremona. ‘The L-functions and modular forms database project’. In: Foundations of Compu-
tutational Mathematics 16.6 (2016), pp. 1541–1553. DOI: 10/ggck7d.

[82] S. Cynk and D. van Straten. ‘Periods of rigid double octic Calabi-Yau threefolds’. In: Annales

Polonici Mathematici 123.1 (2019), pp. 243–258. DOI: 10/ggck7s.

[83] B. Deconinck and M. van Hoeij. ‘Computing Riemann matrices of algebraic curves’. In: Physica
D: Nonlinear Phenomena. Advances in Nonlinear Mathematics and Science 152-153 (15th May
2001). Special issue to honor Vladimir Zakharov, pp. 28–46. DOI: 10/c95vnb.

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[84]

J. van Der Hoeven and G. Lecerf. ‘Composition modulo powers of polynomials’. In: Proceedings
of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation. 2017,
pp. 445–452.

[85] K. Dilcher and K. B. Stolarsky. ‘A polynomial analogue to the Stern sequence’. In: International

Journal of Number Theory 3.01 (2007), pp. 85–103.

[86] A. Dimca. ‘On the de Rham cohomology of a hypersurface complement’. In: American Journal of

Mathematics 113.4 (1991), pp. 763–771. DOI: 10/cw7gpj.

[87] T. Dreyfus, C. Hardouin and J. Roques. ‘Hypertranscendance of solutions of Mahler equations’. In:

Journal of the European Mathematical Society 20 (2018), pp. 2209–2238.

[88] T. Dreyfus, C. Hardouin, J. Roques and M. F. Singer. ‘On the nature of the generating series of walks
in the quarter plane’. In: Inventiones Mathematicae 213.1 (2018), pp. 139–203. DOI: 10.1007/s00
222-018-0787-z.

[89] A.-S. Elsenhans and J. Jahnel. Real and complex multiplication on K3 surfaces via period integration.

27th Feb. 2018. arXiv: 1802.10210.

[90]

[91]

[92]

[93]

J.-C. Faugère and C. Mou. ‘Fast algorithm for change of ordering of zero-dimensional Gröbner
bases with sparse multiplication matrices’. In: ISSAC ’11. Conference proceedings. 2011, pp. 115–
122.

J.-C. Faugère and C. Mou. ‘Sparse FGLM algorithms’. In: Journal of Symbolic Computation 80.3
(2017), pp. 538–569. DOI: 10.1016/j.jsc.2016.07.025.

J.-C. Faugère. ‘A new efficient algorithm for computing Gröbner bases (F4)’. In: Journal of Pure
and Applied Algebra 139.1-3 (1999), pp. 61–88. DOI: 10/bpq5dx.

J.-C. Faugère. ‘A new efficient algorithm for computing Gröbner bases without reduction to zero
(F5)’. In: ISSAC ’02. Conference proceedings. ACM, 2002, pp. 75–83.

[94] G. Fayolle, S. Franceschi and K. Raschel. ‘On the stationary distribution of reflected Brownian
motion in a non-convex wedge’. In: Markov Processes And Related Fields. Markov Processes ans
Related Fields 28.5 (Dec. 2022). 32 pages, 39 ref., p. 32. URL: https://hal.science/hal-03150
317.

[95] G. Fayolle, R. Iasnogorodski and V. Malyshev. Random walks in the quarter plane. Second. Vol. 40.
Probability Theory and Stochastic Modelling. Springer, 2017, pp. xvii+248. DOI: 10.1007/978-3-
319-50930-3. URL: https://doi.org/10.1007/978-3-319-50930-3.

[96] G. Fayolle and P. Mühlethaler. ‘A Markovian analysis of IEEE 802.11 broadcast transmission
networks with buffering’. In: Probab. Engrg. Inform. Sci. 30.3 (2016), pp. 326–344. DOI: 10.1017
/S0269964816000036. URL: https://doi.org/10.1017/S0269964816000036.

[97]

S. Fischler and T. Rivoal. Effective algebraic independence of values of E-functions. Preprint arXiv.
2019.

[98] S. Fischler and T. Rivoal. ‘Approximants de Padé et séries hypergéométriques équilibrées’. In:
Journal de Mathématiques Pures et Appliquées 82.10 (2003), pp. 1369–1394. DOI: 10/d7k2f2.

[99]

J. von zur Gathen and J. Gerhard. Modern computer algebra. Third. Cambridge University Press,
Cambridge, 2013, pp. xiv+795. DOI: 10.1017/CBO9781139856065. URL: https://doi.org/10
.1017/CBO9781139856065.

[100] P. Görlach, C. Lehn and A.-L. Sattelberger. ‘Algebraic analysis of the hypergeometric function
1F1 of a matrix argument’. In: Beiträge zur Algebra und Geometrie 62.2 (2021), pp. 397–427. DOI:
10.1007/s13366-020-00546-z.

[101] D. Y. Grigor’ev. ‘Complexity of factoring and calculating the GCD of linear ordinary differential
operators’. In: Journal of Symbolic Computation 10.1 (1990), pp. 7–37. DOI: 10.1016/S0747-717
1(08)80034-X.

[102] A. J. Guttmann. ‘On the recurrence relation method of series analysis’. In: Journal of Physics. A 8.7

(1975), pp. 1081–1088.

Project MATHEXP

39

[103] A. J. Guttmann and G. S. Joyce. ‘On a new method of series analysis in lattice statistics’. In: Journal
of Physics A: General Physics 5.9 (1972), pp. 81–84. DOI: 10.1088/0305-4470/5/9/001.

[104] C. Hardouin and M. F. Singer. ‘Differential Galois theory of linear difference equations’. In: Math-
ematische Annalen 342.2 (2008), pp. 333–377. DOI: 10.1007/s00208-008-0238-z. URL: http:
//dx.doi.org/10.1007/s00208-008-0238-z.

[105]

J. M. Henn. ‘Lectures on differential equations for Feynman integrals’. In: Journal of Physics A:
Mathematical and Theoretical 48.15 (2015), p. 153001. DOI: 10/ghrjqc.

[106] D. Henrion, J. B. Lasserre and C. Savorgnan. ‘Approximate volume and integration for basic

semialgebraic sets’. In: SIAM Review 51.4 (2009), pp. 722–743. DOI: 10/d3rgrx.

[107] C. Hermite. ‘Sur la fonction exponentielle’. In: Comptes Rendus de l’Académie des Sciences 77

(1873), pp. 18–24.

[108]

[109]

J. van der Hoeven. ‘Fast evaluation of holonomic functions’. In: Theoretical Computer Science
210.1 (1999), pp. 199–215. DOI: 10/b95scc.

J. van der Hoeven. ‘Constructing reductions for creative telescoping’. In: Applicable Algebra in
Engineering, Communication and Computing (2020). DOI: 10.1007/s00200-020-00413-3.

[110] D. L. Hunter and G. A. Baker Jr. ‘Methods of series analysis III. Integral approximant methods’. In:

Physical Review B 19.7 (1979), pp. 3808–3821. DOI: 10.1103/physrevb.19.3808.

[111] A. M. Jasour, N. S. Aybat and C. M. Lagoa. ‘Semidefinite programming for chance constrained
optimization over semialgebraic sets’. In: SIAM Journal on Optimization 25.3 (2015), pp. 1411–
1440. DOI: 10/f7tvft.

[112] A. M. Jasour, A. Hofmann and B. C. Williams. ‘Moment-sum-of-squares approach for fast risk
estimation in uncertain environments’. In: 2018 IEEE Conference on Decision and Control (CDC).
2018, pp. 2445–2451. DOI: 10/ggmx8h.

[113] M. A. H. Khan. ‘High-order differential approximants’. In: Journal of Computational and Applied

Mathematics 149.2 (2002), pp. 457–468. DOI: 10.1016/S0377-0427(02)00561-7.

[114] D. E. Knuth. The art of computer programming. Vol. 2. Addison-Wesley, Reading, MA, 1998.

[115] C. Koutschan, M. Kauers and D. Zeilberger. ‘Proof of George Andrews’s and David Robbins’s
q-TSPP conjecture’. In: Proceedings of the National Academy of Sciences of the USA 108.6 (2011),
pp. 2196–2199. DOI: 10.1073/pnas.1019186108.

[116] C. Koutschan and T. Thanatipanonda. ‘Advanced computer algebra for determinants’. In: Annals

of Mathematics 17.3 (2013), pp. 509–523. DOI: 10.1007/s00026-013-0183-8.

[117] P. Lairez. ‘Computing periods of rational integrals’. In: Mathematics of Computation 85.300 (2016),

pp. 1719–1752. DOI: 10/ggck95.

[118] P. Lairez, M. Mezzarobba and M. Safey El Din. ‘Computing the volume of compact semi-algebraic
sets’. In: ISSAC ’19. Conference proceedings. ACM, 2019, pp. 259–266. DOI: 10/ggck7w.

[119] P. Lairez and E. C. Sertöz. ‘A numerical transcendental method in algebraic geometry: computation
of Picard groups and related invariants’. In: SIAM Journal on Applied Algebra and Geometry 3.4
(2019), pp. 559–584. DOI: 10/ggck6n.

[120]

[121]

[122]

[123]

J. B. Lasserre. ‘Connecting optimization with spectral analysis of tri-diagonal matrices’. 16th Mar.
2020. arXiv: 1907.09784.

J. B. Lasserre. Moments, positive polynomials and their applications. Vol. 1. Series on Optimization
and its Applications. Imperial College Press, Oct. 2009, pp. xxii+361. DOI: 10/ft9x.

J. B. Lasserre. ‘Volume of sublevel sets of homogeneous polynomials’. In: SIAM Journal on Applied
Algebra and Geometry 3.2 (2019), pp. 372–389. DOI: 10/ggck69.

J. B. Lasserre, V. Magron, S. Marx and O. Zahm. ‘Minimizing rational functions: a hierarchy of
approximations via pushforward measures’. 10th Dec. 2020. arXiv: 2012.05793.

[124] L. F. Matusevich. ‘Weyl closure of hypergeometric systems’. In: Collectanea Mathematica 60.2

(June 2009), pp. 147–158. DOI: 10.1007/bf03191207.

40

Inria Annual Report 2023

[125] M. Mezzarobba. ‘NumGfun: a package for numerical and analytic computation and D-finite
functions’. In: ISSAC ’10. Ed. by S. M. Watt. Conference proceedings. ACM, 2010, pp. 139–146. DOI:
10/cg7w72.

[126] M. Mezzarobba. Rigorous multiple-precision evaluation of D-finite functions in Sagemath. 7th July

2016. arXiv: 1607.01967.

[127] P. Molin and C. Neurohr. ‘Computing period matrices and the Abel-Jacobi map of superelliptic

curves’. In: Mathematics of Computation 88.316 (2019), pp. 847–888. DOI: 10/ggck8t.

[128]

S. Naldi and V. Neiger. ‘A divide-and-conquer algorithm for computing Gröbner bases of syzygies
in finite dimension’. In: ISSAC ’20. Conference proceedings. 2020, pp. 380–387.

[129] C. Neurohr. ‘Efficient integration on Riemann surfaces and applications’. Ph.D. Thesis. Universität

Oldenburg, 2018.

[130] T. Oaku. ‘Algorithms for b-functions, restrictions, and algebraic local cohomology groups of

D-modules’. In: Advances in Applied Mathematics 19.1 (1997), pp. 61–105. DOI: 10/dzvhg3.

[131] T. Oaku and N. Takayama. ‘Algorithms for D-modules: restriction, tensor product, localization,
and local cohomology groups’. In: Journal of Pure and Applied Algebra 156.2-3 (2001), pp. 267–308.
DOI: 10/bct97n.

[132] R. Pagès. ‘Factoring differential operators over algebraic curves in positive characteristic’. In:
ISSAC 2022 - International Symposium on Symbolic and Algebraic Computation. Lille, France,
20th May 2022. URL: https://hal.science/hal-03759105.

[133] K. Pardue. Nonstandard Borel-fixed ideals. Brandeis University, 1994.

[134] M. Petkovšek. ‘Hypergeometric solutions of linear recurrences with polynomial coefficients’. In:

Journal of Symbolic Computation 14 (1992), pp. 243–264.

[135] G. Pólya. Mathematics and plausible reasoning. Induction and analogy in mathematics. Princeton

U. Press, 1954, pp. xvi+280.

[136] G. Pólya. How to solve it. A new aspect of mathematical method. Princeton U. Press, 1945,

pp. xxviii+253.

[137] D. Popescu. ‘General Néron desingularization and approximation’. In: Nagoya Mathematical
Journal 104 (1986), pp. 85–115. DOI: 10.1017/S0027763000022698. URL: https://doi.org/1
0.1017/S0027763000022698.

[138] M. van der Put. ‘Differential equations in characteristic p’. In: vol. 97. 1-2. Special issue in honour
of Frans Oort. 1995, pp. 227–251. URL: http://www.numdam.org/item?id=CM_1995__97_1-2
_227_0.

[139]

J. Roques. ‘On the algebraic relations between Mahler functions’. In: Transactions of the AMS 370.1
(2018), pp. 321–355.

[140] E. C. Sertöz. ‘Computing periods of hypersurfaces’. In: Mathematics of Computation 88.320 (2019),

pp. 2987–3022. DOI: 10/ggck7t.

[141] A. B. Šidlovski˘ı. ‘On transcendentality of the values of a class of entire functions satisfying linear

differential equations’. In: Doklady Akademii Nauk SSSR 105 (1955), pp. 35–37.

[142] C. L. Siegel. ‘Über einige Anwendungen diophantischer Approximationen [reprint of Abhandlun-
gen der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 1929, Nr.
1]’. In: On some applications of Diophantine approximations. Vol. 2. Quad./Monogr. Ed. Norm.,
Pisa, 2014, pp. 81–138.

[143] M. F. Singer. ‘Algebraic solutions of nth order linear differential equations’. In: Proceedings of the
Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979). Vol. 54. Queen’s Papers in Pure
and Appl. Math. Queen’s Univ., Kingston, Ont., 1980, pp. 379–420.

[144] L. Slot and M. Laurent. ‘Near-optimal analysis of Lasserre’s univariate measure-based bounds for
multivariate polynomial optimization’. In: Mathematical Programming (2020). DOI: 10/ghzg5h.

Project MATHEXP

41

[145] A. V. Smirnov and M. N. Tentyukov. ‘Feynman integral evaluation by a sector decomposition
approach (FIESTA)’. In: Computer Physics Communications 180.5 (2009), pp. 735–746. DOI: 10/c9
2sm7.

[146] V. N. Sorokin. ‘A transcendence measure for π2’. In: Sbornik: Mathematics 187.12 (1996), pp. 1819–

1852.

[147] R. P. Stanley. ‘Differentiably finite power series’. In: European Journal of Combinatorics 1.2 (1980),

pp. 175–188. DOI: 10.1016/S0195-6698(80)80051-5.

[148] H. Tsai. ‘Algorithms for associated primes, Weyl closure, and local cohomology of D-modules’. In:
Local cohomology and its applications (Guanajuato, 1999). Vol. 226. Lecture Notes in Pure and
Appl. Math. Dekker, 2002, pp. 169–194.

[149] H. Tsai. ‘Weyl closure of a linear differential operator’. In: Journal of Symbolic Computation 29.4-5
(2000). Special issue on Symbolic Computation in Algebra, Analysis, and Geometry (Berkeley, CA,
1998), pp. 747–775. DOI: 10.1006/jsco.1999.0400. URL: http://dx.doi.org/10.1006/jsco
.1999.0400.

[150] W. T. Tutte. ‘Chromatic sums for rooted planar triangulations: the cases λ = 1 and λ = 2’. In:
Canadian Journal of Mathematics 25 (1973), pp. 426–447. DOI: 10.4153/CJM-1973-043-3. URL:
https://doi.org/10.4153/CJM-1973-043-3.

[151] D. Zagier. ‘The arithmetic and topology of differential equations’. In: European Congress of Math-

ematics. European Mathematical Society, Zürich, 2018, pp. 717–776.

[152] D. Zeilberger and W. Zudilin. ‘The irrationality measure of π is at most 7.103205334137. . . ’. In:
Moscow Journal of Combinatorics and Number Theory 9.4 (2020), pp. 407–419. DOI: 10/gjnbf8.
<scratchpad>
- Main topic: Research Centre Inria Saclay Centre
- Key points:
  - The team develops both experimental mathematics and computer algebra in parallel.
  - They aim to discover and prove new mathematical results using computer tools.
  - The team focuses on four interconnected categories: algebraic algorithms, symbolic integration, computerized classification, and applications in diverse areas.
  - They have made significant progress in various areas, including algebraic solutions of linear differential equations, algebraic classification of functions and numbers, and efficient resolution of functional equations.
- Structure: The document is organized into sections, including "Activity Report," "Project-Team MATHEXP," "Domain," "Theme," "Research Program," "Application Domains," and "Partnerships and Cooperations."
- Potential artifacts to ignore: Web-related elements, such as links, HTML tags, and other web-related elements.
</scratchpad>

<final_summary>
The Inria Saclay Centre has been actively involved in research activities, focusing on experimental mathematics and computer algebra. The team has developed various algorithms and techniques, including algebraic solutions of linear differential equations, algebraic classification of functions and numbers, and efficient resolution of functional equations. They have also made significant progress in applications in diverse areas, such as combinatorics, probability theory, number theory, algebraic geometry, statistical physics, and quantum mechanics. The team's research has been recognized through various initiatives, including the ERC Starting Grant "10000 DIGITS" and the ANR project De rerum natura.
The Inria Saclay Centre has been actively involved in research activities, focusing on experimental mathematics and computer algebra. The team has developed various algorithms and techniques, including algebraic solutions of linear differential equations, algebraic classification of functions and numbers, and efficient resolution of functional equations. They have also made significant progress in applications in diverse areas, such as combinatorics, probability theory, number theory, algebraic geometry, statistical physics, and quantum mechanics. The team's research has been recognized through various initiatives, including the ERC Starting Grant "10000 DIGITS" and the ANR project De rerum natura.