2023Activity reportProject-TeamMATHEXP

2.1 Experimental mathematics

In science, observation and experiment play an important role in formulating hypotheses. In mathematics, 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  55, 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 computer 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, 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 ${\int }_0^{1}f(x,y)\mathrm{d}y$? 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 applications 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

whose analysis leads to the best known measure of irrationality of $\pi$, 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 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 obtained 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 experimental mathematics. We target especially four tasks of general interest: algebraic algorithms for manipulating 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.

3.1 Algebraic algorithms for multivariate systems of equations

At large, MATHEXP deals with algebraic and seminumerical methods. This part goes through the fundamental 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.

3.2 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 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 $𝐭=t_1,\cdots ,t_s$ and $𝐱=x_1,\cdots ,x_n$, and an integration domain $\Omega \left(𝐭\right)\subseteq {ℝ}^n$, how to compute the function

Concretely, $F(𝐭,𝐱)$ is described by a system of linear PDEs with polynomial coefficients, $\Omega \left(𝐭\right)$ is given by polynomial inequalities, and we want a system of PDEs describing $G\left(𝐭\right)$. 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 development. 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.

3.3 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.

3.4 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 60115, 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  67 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  102, 103, 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  43; 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.

3.5 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 seminumerical 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  40. 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 conviction. A celebrated example is the experimental discovery of the BBP formula for $\pi$  41 (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.

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 k 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.

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

In 18, Notarantonio and Yurkevich studied systems of $n\ge 1$ discrete differential equations of order $k\ge 1$ 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 solutions

In 32, Notarantonio and Yurkevich studied systems of $n\ge 1$, not necessarily linear, discrete differential equations (DDEs) of order $k\ge 1$ 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 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 $P^n\left(x\right)Q\left(x\right)$, where $P\left(x\right)$ and $Q\left(x\right)$ 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 $𝐏\subset {ℝ}^3$ 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.

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

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial $N!$ in arithmetic complexity quasi-linear in $\sqrt{N}$. 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 $\sqrt{N}$; 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.

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 probability 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 formulae 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 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 $\mathbb{Q}\left(x\right)$, 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 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 polynomial 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 $2^n-\left(\genfrac{}{}{0pt}{}{n+1}{⌊\frac{n+1}{2}⌋}\right)$ supporting the conjecture that the sunset Feynman integrals 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 $K3$ 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 79 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.

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

Creative telescoping is an algorithmic method introduced by Zeilberger in the 1990s to compute parametrized 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 telescoping 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 lexicographic Gröbner basis $𝒢$ of an ideal in $\mathbb{Q}[x,y]$ with a generating set in $\mathbb{Z}[x,y]$. The algorithm shows a complexity that is less than cubic in terms of the dimension of $\mathbb{Q}[x,y]/\langle 𝒢\rangle$ 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 $I\subseteq 𝕂[x,y]$, 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 $I\subseteq 𝕂[x,y]$ be an ideal generated by $ℱ$, a subset of $𝔸[x,y]$ with $𝔸\hookrightarrow 𝕂$ 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 $𝔪\subseteq 𝔸$ is a good maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca and Valla 33, 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 $𝔪\subseteq 𝔸$ to build the $𝔪$-adic expansion of $𝒢$. Inspired by Pardue 133, a constructive proof is given to characterise a Zariski open set of ${\mathrm{GL}}_2\left(𝕂\right)$ (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 $\left|𝕂\right|$ is sufficiently large. This sharpened her analysis to obtain, when $𝔸=\mathbb{Z}$ or $𝔸=k\left[t\right]$, a complexity less than cubic in terms of the dimension of $\mathbb{Q}[x,y]/\langle 𝒢\rangle$ and softly linear in the height of the coefficients of $𝒢$. The resulting method is also adapted to extract $\langle x,y\rangle$-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.

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 ${\mathbb{P}}^3$ with six $A_1$ 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).

7.1 International initiatives

7.2 European initiatives

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.)
• $\partial$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).

8.1 Promoting scientific activities

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 Université, France.
  • Raphaël Pagès, Mathématiques générales and Outils Mathématiques, 90h, L1, Université 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 Polytechnique, 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.

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.

9.1 Publications of the year

9.2 Cited publications

• 33 articleA.A. Conca and G.G. Valla. Canonical Hilbert-Burch matrices for ideals of $k[x,y]$.Michigan Mathematical Journal572008, 157 -- 172URL: https://doi.org/10.1307/mmj/1220879402DOIback to text
• 34 articleS. A.S. A. Abramov, M. A.M. A. Barkatou and M.M. van Hoeij. Apparent singularities of linear difference equations with polynomial coefficients.1722006, 117--133DOIback to text
• 35 inproceedingsS. A.Sergei A. Abramov and M.Mark van Hoeij. Desingularization of linear difference operators with polynomial coefficients.ISSAC~'99Conference proceedingsACM1999DOIback to text
• 36 articleB.Boris Adamczewski and C.Colin Faverjon. Méthode de Mahler, transcendance et relations linéaires: aspects effectifs.3022018, 557--573back to text
• 37 articleB.Boris Adamczewski and T.Tanguy Rivoal. Exceptional values of $E$-functions at algebraic points.5042018, 697--908back to textback to text
• 38 articleM. F.Michael F. Adamer, A. C.András C. Lőrincz, A.-L.Anna-Laura Sattelberger and B.Bernd Sturmfels. Algebraic analysis of rotation data.112December 2020, 189--211DOIback to text
• 39 reportS.S. Badger, J.J. Bendavid, V.V. Ciulli, A.A. Denner, R.R. Frederix, M.M. Grazzini, J.J. Huston, M.M. Schönherr, K.K. Tackmann, J.J. Thaler, C.C. Williams, J. R.J. R. Andersen, K.K. Becker, M.M. Bell, J.J. Bellm, E.E. Bothmann, R.R. Boughezal, J.J. Butterworth, S.S. Carrazza, M.M. Chiesa, L.L. Cieri, M.M. Duehrssen-Debling, G.G. Falmagne, S.S. Forte, P.P. Francavilla, M.M. Freytsis, J.J. Gao, P.P. Gras, N.N. Greiner, D.D. Grellscheid, G.G. Heinrich, G.G. Hesketh, S.S. Höche, L.L. Hofer, T.-J.T.-J. Hou, A.A. Huss, J.J. Isaacson, A.A. Jueid, S.S. Kallweit, D.D. Kar, Z.Z. Kassabov, V.V. Konstantinides, F.F. Krauss, S.S. Kuttimalai, A.A. Lazapoulos, P.P. Lenzi, Y.Y. Li, J. M.J. M. Lindert, X.X. Liu, G.G. Luisoni, L.L. Lönnblad, P.P. Maierhöfer, D.D. Maître, A. C.A. C. Marini, G.G. Montagna, M.M. Moretti, P. M.P. M. Nadolsky, G.G. Nail, D.D. Napoletano, O.O. Nicrosini, C.C. Oleari, D.D. Pagani, C.C. Pandini, L.L. Perrozzi, F.F. Petriello, F.F. Piccinini, S.S. Plätzer, I.I. Pogrebnyak, S.S. Pozzorini, S.S. Prestel, C.C. Reuschle, J.J. Rojo, L.L. Russo, P.P. Schichtel, S.S. Schumann, A.A. Siódmok, P.P. Skands, D.D. Soper, G.G. Soyez, P.P. Sun, F. J.F. J. Tackmann, E.E. Takasugi, S.S. Uccirati, U.U. Utku, L.L. Viliani, E.E. Vryonidou, B. T.B. T. Wang, B.B. Waugh, M. A.M. A. Weber, J.J. Winter, K. P.K. P. Xie, C.-P.C.-P. Yuan, F.F. Yuan, K.K. Zapp and M.M. Zaro. Les Houches 2015: Physics at TeV colliders standard model working group report.May 2016back to text
• 40 articleD.David Bailey and J. M.J. M. Borwein. High-precision numerical integration: progress and challenges.4672011, 741--754DOIback to textback to text
• 41 articleD.David Bailey, P.Peter Borwein and S.Simon Plouffe. On the rapid computation of various polylogarithmic constants.662181997, 903--913DOIback to text
• 42 inproceedingsM.Moulay Barkatou, T.Thomas Cluzeau, L.Lucia Di Vizio and J.-A.Jacques-Arthur Weil. Computing the Lie algebra of the differential Galois group of a linear differential system.ISSAC~'2016Conference proceedingsACM2016, 63--70back to text
• 43 articleB.Bernhard Beckermann and G.George Labahn. A uniform approach for the fast computation of matrix-type Padé approximants.1531994, 804--823DOIback to textback to text
• 44 articleJ. P.Jason P. Bell and M.Michael Coons. Transcendence tests for Mahler functions.14532017, 1061--1070back to text
• 45 articleO.Olivier Bernardi and M.Mireille Bousquet-Mélou. Counting colored planar maps: algebraicity results.10152011, 315--377URL: https://doi.org/10.1016/j.jctb.2011.02.003DOIback to textback to text
• 46 articleO.Olivier Bernardi, M.Mireille Bousquet-Mélou and K.Kilian Raschel. Counting quadrant walks via Tutte's invariant method.12021DOIback to text
• 47 inproceedingsJ.Jérémy Berthomieu, C.Christian Eder and M.Mohab Safey El Din. Msolve: a library for solving polynomial systems.ISSAC~'21Conference proceedingsACMJuly 2021, 51--58DOIback to textback to text
• 48 inproceedingsJ.Jérémy Berthomieu and J.-C.Jean-Charles Faugère. A polynomial-division-based algorithm for computing linear recurrence relations.ISSAC~'18Conference proceedings2018, 79--86back to text
• 49 articleD.Daniel Bertrand and F.Frits Beukers. Équations différentielles linéaires et majorations de multiplicités.1811985, 181--192back to text
• 50 articleF.F. Beukers. A note on the irrationality of $\left(2\right)$ and $\left(3\right)$.1131979, 268--272back to text
• 51 articleF.F. Beukers. A refined version of the Siegel-Shidlovskii theorem.16312006, 369--379URL: https://doi.org/10.4007/annals.2006.163.369DOIback to text
• 52 incollectionF.F. Beukers. Padé-approximations in number theory.Padé Approximation and its Applications, Amsterdam 1980888Lecture Notes in Math.Springer1981, 90--99back to text
• 53 articleS.Spencer Bloch, M.Matt Kerr and P.Pierre Vanhove. A Feynman integral via higher normal functions.151122015, 2329--2375DOIback to textback to text
• 54 articleS.Spencer Bloch and P.Pierre Vanhove. The elliptic dilogarithm for the sunset graph.1482015, 328--364DOIback to text
• 55 bookJ.Jonathan Borwein and D.David Bailey. Mathematics by experiment.Plausible reasoning in the 21st centuryA K Peters2008, xii+377back to text
• 56 bookA.A. Bostan, F.F. Chyzak, M.M. Giusti, R.R. Lebreton, G.G. Lecerf, B.B. Salvy and É.É. Schost. Algorithmes efficaces en calcul formel.686 pagesCreateSpace2017back to text
• 57 inproceedingsA.Alin Bostan, F.Frédéric Chyzak, P.Pierre Lairez and B.Bruno Salvy. Generalized Hermite reduction, creative telescoping and definite integration of D-finite functions.ISSAC~'18Conference proceedingsACM2018, 95--102DOIback to text
• 58 inproceedingsA.Alin Bostan, F.Frédéric Chyzak, G.Grégoire Lecerf, B.Bruno Salvy and É.Éric Schost. Differential equations for algebraic functions.ISSAC~'07Conference proceedings2007, 25--32back to text
• 59 articleA.A. Bostan, C.-P.C.-P. Jeannerod, C.C. Mouilleron and É.É. Schost. On matrices with displacement structure: generalized operators and faster algorithms.3832017, 733--775URL: https://doi.org/10.1137/16M1062855DOIback to text
• 60 articleA.A. Bostan and M.M. Kauers. The complete generating function for Gessel walks is algebraic.1389With an appendix by Mark van Hoeij2010, 3063--3078DOIback to textback to text
• 61 inproceedingsA.Alin Bostan, P.Pierre Lairez and B.Bruno Salvy. Creative telescoping for rational functions using the Griffiths–Dwork method.ISSAC~'13Conference proceedingsACM2013, 93--100DOIback to text
• 62 articleA.Alin Bostan, P.Pierre Lairez and B.Bruno Salvy. Multiple binomial sums.80part 22017, 351--386URL: https://doi.org/10.1016/j.jsc.2016.04.002DOIback to text
• 63 articleA.Alin Bostan, T.Tanguy Rivoal and B.Bruno Salvy. Explicit degree bounds for right factors of linear differential operators.5312020, 53--62DOIback to textback to text
• 64 articleM.Mireille Bousquet-Mélou and A.Arnaud Jehanne. Polynomial equations with one catalytic variable, algebraic series and map enumeration.9652006, 623--672back to textback to textback to textback to text
• 65 articleM.Mireille Bousquet-Mélou. Square lattice walks avoiding a quadrant.1442016, 37--79back to text
• 66 inproceedingsB.Brice Boyer, C.Christian Eder, J.-C.Jean-Charles Faugère, S.Sylvian Lachartre and F.Fayssal Martani. GBLA: Gröbner basis linear algebra package.ISSAC~'16Conference proceedingsACM2016, 135--142DOIback to text
• 67 articleR.R. Brak and A. J.A. J. Guttmann. Algebraic approximants: a new method of series analysis.23241990, L1331--L1337DOIback to text
• 68 articleN.Nils Bruin, J.Jeroen Sijsling and A.Alexandre Zotine. Numerical computation of endomorphism rings of Jacobians.2113th Algorithmic Number Theory Symposium2019, 155--171DOIback to text
• 69 inproceedingsM.Manfred Buchacher, M.Manuel Kauers and G.Gleb Pogudin. Separating variables in bivariate polynomial ideals.ISSAC~'20Conference proceedingsACM2020, 54--61DOIback to text
• 70 bookJ. C.J. C. Butcher. Numerical methods for ordinary differential equations.John Wiley & Sons2016, xxiii+513DOIback to text
• 71 incollectionS.Shaoshi Chen, M.Maximilian Jaroschek, M.Manuel Kauers and M. F.Michael F. Singer. Desingularization explains order-degree curves for Ore operators.ISSAC~'13Conference proceedingsACM2013, 157--164back to text
• 72 articleS.Shaoshi Chen, M.Manuel Kauers, Z.Ziming Li and Y.Yi Zhang. Apparent singularities of D-finite systems.952019, 217--237DOIback to text
• 73 incollectionS.Shaoshi Chen and M.Manuel Kauers. Order-degree curves for hypergeometric creative telescoping.ISSAC~'12Conference proceedingsACM2012, 122--129back to text
• 74 articleS.Shaoshi Chen and M.Manuel Kauers. Trading order for degree in creative telescoping.4782012, 968--995DOIback to text
• 75 incollectionD. V.David V. Chudnovsky and G. V.Gregory V. Chudnovsky. Computer algebra in the service of mathematical physics and number theory.Computers in Mathematics (Stanford, CA, 1986)125Lecture Notes in Pure and Appl. Math.Dekker1990, 109--232back to text
• 76 inproceedingsD. V.David V. Chudnovsky and G. V.Gregory V. Chudnovsky. Computer assisted number theory with applications.Number theory (New York, 1984--1985)1240Lecture Notes in MathematicsSpringer1987, 1--68DOIback to text
• 77 articleF.Frédéric Chyzak, T.Thomas Dreyfus, P.Philippe Dumas and M.Marc Mezzarobba. Computing solutions of linear Mahler equations.873142018, 2977--3021back to textback to text
• 78 inproceedingsF.Frédéric Chyzak and P.Philippe Dumas. A Gröbner-basis theory for divide-and-conquer recurrences.ISSAC~'20Conference proceedingsACMJuly 2020DOIback to text
• 79 inproceedingsT.Thomas Cluzeau. Factorization of differential systems in characteristic $p$.Proceedings of the 2003 International Symposium on Symbolic and Algebraic ComputationACM, New York2003, 58--65URL: https://doi.org/10.1145/860854.860875DOIback to text
• 80 articleE.Edgar Costa, N.Nicolas Mascot, J.Jeroen Sijsling and J.John Voight. Rigorous computation of the endomorphism ring of a Jacobian.883172019, 1303--1339DOIback to text
• 81 articleJ.John Cremona. The L-functions and modular forms database project.1662016, 1541--1553DOIback to text
• 82 articleS.Sławomir Cynk and D.Duco van Straten. Periods of rigid double octic Calabi-Yau threefolds.12312019, 243--258DOIback to textback to text
• 83 articleB.Bernard Deconinck and M.Mark van Hoeij. Computing Riemann matrices of algebraic curves.152-153Special issue to honor Vladimir ZakharovMay 2001, 28--46DOIback to text
• 84 inproceedingsJ.Joris van Der Hoeven and G.Grégoire Lecerf. Composition modulo powers of polynomials.Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation2017, 445--452back to text
• 85 articleK.Karl Dilcher and K. B.Kenneth B Stolarsky. A polynomial analogue to the Stern sequence.3012007, 85--103back to text
• 86 articleA.Alexandru Dimca. On the de Rham cohomology of a hypersurface complement.11341991, 763--771DOIback to text
• 87 articleT.Thomas Dreyfus, C.Charlotte Hardouin and J.Julien Roques. Hypertranscendance of solutions of Mahler equations.202018, 2209--2238back to text
• 88 articleT.Thomas Dreyfus, C.Charlotte Hardouin, J.Julien Roques and M. F.Michael F. Singer. On the nature of the generating series of walks in the quarter plane.21312018, 139--203DOIback to text
• 89 miscA.-S.Andreas-Stephan Elsenhans and J.Jörg Jahnel. Real and complex multiplication on K3 surfaces via period integration.February 2018back to textback to text
• 90 articleJ.-C.Jean-Charles Faugère. A new efficient algorithm for computing Gröbner bases $\left(F_4\right)$.1391-3Effective methods in algebraic geometry (Saint-Malo, 1998)1999, 61--88DOIback to text
• 91 inproceedingsJ.-C.Jean-Charles Faugère. A new efficient algorithm for computing Gröbner bases without reduction to zero $\left(F_5\right)$.ISSAC~'02Conference proceedingsACM2002, 75--83back to text
• 92 inproceedingsJ.-C.J.-Ch. Faugère and C.Ch. Mou. Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices.ISSAC~'11Conference proceedings2011, 115--122back to text
• 93 articleJ.-C.J.-Ch. Faugère and C.Ch. Mou. Sparse FGLM algorithms.8032017, 538--569DOIback to text
• 94 articleG.Guy Fayolle, S.Sandro Franceschi and K.Kilian Raschel. On the stationary distribution of reflected Brownian motion in a non-convex wedge.Markov Processes And Related Fields28532 pages, 39 ref.December 2022, 32HALback to text
• 95 bookG.Guy Fayolle, R.Roudolf Iasnogorodski and V.Vadim Malyshev. Random walks in the quarter plane.40Probability Theory and Stochastic ModellingSpringer2017, xvii+248URL: https://doi.org/10.1007/978-3-319-50930-3DOIback to text
• 96 articleG.Guy Fayolle and P.Paul Mühlethaler. A Markovian analysis of IEEE 802.11 broadcast transmission networks with buffering.Probab. Engrg. Inform. Sci.3032016, 326--344URL: https://doi.org/10.1017/S0269964816000036DOIback to text
• 97 articleS.Stéphane Fischler and T.Tanguy Rivoal. Approximants de Padé et séries hypergéométriques équilibrées.82102003, 1369--1394DOIback to text
• 98 miscS.S. Fischler and T.T. Rivoal. Effective algebraic independence of values of E-functions.Preprint arXiv2019back to text
• 99 bookJ.Joachim von zur Gathen and J.Jürgen Gerhard. Modern computer algebra.Cambridge University Press, Cambridge2013, xiv+795URL: https://doi.org/10.1017/CBO9781139856065DOIback to text
• 100 articleP.Paul Görlach, C.Christian Lehn and A.-L.Anna-Laura Sattelberger. Algebraic analysis of the hypergeometric function ${}_1{F}_1$ of a matrix argument.6222021, 397--427DOIback to text
• 101 articleD. Y.D. Yu. Grigor'ev. Complexity of factoring and calculating the GCD of linear ordinary differential operators.1011990, 7--37DOIback to text
• 102 articleA. J.A. J. Guttmann and G. S.G. S. Joyce. On a new method of series analysis in lattice statistics.591972, 81--84DOIback to text
• 103 articleA. J.A. J. Guttmann. On the recurrence relation method of series analysis.871975, 1081--1088back to text
• 104 articleC.Charlotte Hardouin and M. F.Michael F. Singer. Differential Galois theory of linear difference equations.34222008, 333--377URL: http://dx.doi.org/10.1007/s00208-008-0238-zDOIback to text
• 105 articleJ. M.Johannes M. Henn. Lectures on differential equations for Feynman integrals.48152015, 153001DOIback to text
• 106 articleD.Didier Henrion, J. B.Jean B. Lasserre and C.Carlo Savorgnan. Approximate volume and integration for basic semialgebraic sets.5142009, 722--743DOIback to text
• 107 articleC.Charles Hermite. Sur la fonction exponentielle.771873, 18--24back to text
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