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  45, 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  86. 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  138. 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  121, 120. 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 50102, 103. 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  57 or differential approximation  100, 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  89, 90, 97, 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  33; 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 101, 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  30. 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 $\pi$  31 (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 Institutional life

In the year 2022, the activity of the team MATHEXP was impacted by the bad mood and heavy dysfunctions in the institute.

• The disastrous failure of the deployment of a new financial and human resource information system (Eksae) impacts negatively all administrative colleagues with whom researchers interact. It forces the former to duplicate their works, when not making them unable to perform even simple tasks. It also directly impacts researchers, who cannot oversee their budget in any reliable way, and occasionally have to delay simple purchases as a consequence of a reprioritization of overburdened administrative forces.
• The continuously diminishing emphasis on research in the presentation of Inria and the growing fuzziness around the status of researchers and the role and position of the institute attack the meaning of work and our collective values.
• The extremely degraded relationship between the Inria Commission d'Évaluation and the current Inria direction is unbearable. It causes a general distrust in the direction's intentions and in the outcome of future hiring and promoting process. The team deplores the current situation. It thanks the Commission d'Évaluation for its outstanding communication to the researchers it represents and for its continued effort in organizing and participating in Inria hiring and promotion committees.

5.2 Creation of the project-team

Project-team MATHEXP has been officially created on April 1st, 2022. It evolves from the former SPECFUN project-team, with a more definite bias towards experimental mathematics.

5.3 ERC project: 10,000 DIGITS

Lairez's ERC proposal, which had been retained for funding with a grant of roughly 1.4 million euros, has started its activities, including first hirings. The project focuses on the foundations of transcendental methods in numerical nonlinear algebra.

5.4 David P. Robbins Prize

Alin Bostan, of the team, together with Irina Kurkova and Kilian Raschel, have received the 2022 AMS David P. Robbins Prize for their paper “A human proof of Gessel's lattice path conjecture,” published in Transactions of the American Mathematical Society in 2017. The paper proves highly nontrivial enumeration results on a family of lattice paths known as Gessel walks.

5.5 Promotion

Frédéric Chyzak was nominated “Directeur de Recherche” in 2022.

6.1 Algorithms for discrete differential equations of order 1

Discrete differential equations of order 1 relate in a polynomial fashion a power series $F(t,u)$ in $t$ whose coefficients are polynomial in a “catalytic” variable $u$ to one of its specializations, say $F(t,1)$. Such equations are ubiquitous in combinatorics, notably in the enumeration of maps and walks. When the solution $F$ is unique, a celebrated result by Bousquet-Mélou and Jehanne, reminiscent of Popescu's theorem in commutative algebra, states that $F$ is algebraic. In 12, Alin Bostan, Frédéric Chyzak, and Hadrien Notarantonio, together with Mohab Safey El Din (Sorbonne Université), address algorithmic and complexity questions related to this result. In generic situations, they first revisited and analyzed known algorithms, based either on polynomial elimination or on the guess-and-prove paradigm. They then designed two new algorithms: the first has a geometric flavor, the second blends elimination and guess-and-prove. In the general case (no genericity assumptions), they proved that the total arithmetic size of the algebraic equations for $F(t,1)$ is bounded polynomially in the size of the input discrete differential equation, and that one can compute such equations in polynomial time.

6.2 Symbolic-numeric factorization of differential operators

Frédéric Chyzak and Alexandre Goyer, together with Marc Mezzarobba (LIX), presented in 13 a symbolic-numeric Las Vegas algorithm for factoring Fuchsian ordinary differential operators with rational function coefficients. The new algorithm combines ideas of van Hoeij's ‘‘local-to-global” method and of the ‘‘analytic” approach proposed by van der Hoeven. It essentially reduces to the former in ‘‘easy” cases where the local-to-global method succeeds, and to an optimized variant of the latter in the ‘‘hardest” cases, while handling intermediate cases more efficiently than both.

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

In 21, Hadrien Notarantonio and Sergey Yurkevich studied systems of discrete differential equations of general order in one catalytic variable. They provided a constructive and elementary proof of algebraicity of the solutions of such systems. They obtained effective bounds and developed a systematic method for computing the minimal polynomials. Their approach was a generalization of the pioneering work by Bousquet-Mélou and Jehanne (2006).

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.

1. In 8, he 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. He 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 Stability and string stability of car-following models with reaction-time delay

In 10, Guy Fayolle, in collaboration with J.-M. Lagouttes (Inria Paris) and C. Flores (Institute of Transportation Studies, UCB), have investigated the transfer function emanating from the linearization of a car-following model for human drivers when taking into account their reaction time, which is known to be a cause of the stop-and-go traffic phenomenon. This led to a non rational transfer function, implying nontrivial stability conditions that they could give explicitly. These conditions are in particular satisfied whenever string stability holds. It is also shown how the reaction time can introduce a regime of partial string stability, where the transfer function modulus remains smaller than 1, up to some critical frequency. The authors explored conditions in the parameter space that discriminate between 3 different regimes (stability, string stability, partial string stability).

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 17, 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 Solving Rupert's problem algorithmically

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. The paper 22 by Sergey Yurkevich in collaboration with Jakob Steininger is an extended abstract of their joint work in which they proved that Rupert's problem is algorithmically decidable for polyhedra with algebraic coordinates. There they 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 The art of algorithmic guessing in gfun

The technique of guessing can be very fruitful when dealing with sequences which arise in practice. This holds true especially when guessing is performed algorithmically and efficiently. The ideal tool for it exists as a package named gfun in the software Maple. In the by now published paper 11 Sergey Yurkevich explores and explains some of gfun's possibilities and illustrates them on two examples from recent mathematical research by him and his collaborators.

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 6, 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 On a class of hypergeometric diagonals

In 7, Alin Bostan and Sergey Yurkevich proved that the diagonal of any finite product of algebraic functions of the form

is a generalized hypergeometric function, and they provided explicit description of its parameters. The particular case ${(1-x-y)}^R/(1-x-y-z)$ corresponds to the main identity of Abdelaziz, Koutschan and Maillard in  23. The result in 7 is useful in both directions: on the one hand it shows that Christol's conjecture holds true for a large class of hypergeometric functions, on the other hand it allows for a very explicit and general viewpoint on the diagonals of algebraic functions of the type above. Finally, in contrast to  23, the new proof is completely elementary and does not require any algorithmic help.

6.11 A hypergeometric proof that $\mathrm{𝖨𝗌𝗈}$ is bijective

A short and elementary proof of the main technical result of the recent article “On the uniqueness of Clifford torus with prescribed isoperimetric ratio” 136 by Thomas Yu and Jingmin Chen has been found by Alin Bostan and Sergey Yurkevich in 5. The key of the new proof is an explicit expression of the central function (Iso, proved to be bijective) as a quotient of Gaussian hypergeometric functions.

6.12 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 3, Alin Bostan proposes a shorter and elementary proof.

6.13 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.

6.14 On some combinatorial sequences associated to invariant theory

In 4, Alin Bostan together with Jordan Tirrell (Washington College, USA), Bruce W. Westbury (U. Texas at Dallas, USA) and Yi Zhang (Xi'an Jiaotong-Liverpool University, Suzhou, China) study the enumerative and analytic properties of some sequences constructed using tensor invariant theory. The first family, containing the so-called octant sequences, is constructed from the exceptional Lie group $G_2$. The second family, containing the so-called quadrant sequences, is constructed from the special linear group $SL\left(3\right)$. All sequences are defined as the dimension of the subspace of invariant tensors in the tensor powers of the corresponding representation. The authors first give combinatorial interpretations for the first octant sequence, $T_3$, based on interpretations of the sequences in the first family as lattice walks in the plane. They then show that the second octant sequence $E_3$ is the binomial transform of $T_3$; this result provides an unexpected connection between the invariant theory of $G_2$ and the combinatorics of set partitions. A third result is a proof (actually three independent proofs) of a recurrence satisfied by $T_3$ that was conjectured by Mihailovs in the early 2000s. Similar results are obtained for the sequences of the quadrant sequences. These sequences also have interpretations as enumerating two-dimensional lattice walks. They are all P-recursive, and recurrence relations are proved for them. In all cases the associated differential operators are of third order and have the remarkable property that they can be solved to give closed formulae for the ordinary generating functions in terms of classical Gaussian hypergeometric functions. Moreover, it is shown that the octant sequences and the quadrant sequences are related by the branching rules for the inclusion of $SL\left(3\right)$ in $G_2$.

6.15 Continued fractions, orthogonal polynomials and Dirichlet series

Using an experimental mathematics approach, Alin Bostan together with Frédéric Chapoton (CNRS, IRMA Strasbourg) obtained in 15 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.

6.16 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 16 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.

6.17 Gröbner bases and critical values: the asymptotic combinatorics of determinantal systems

In 2 Alin Bostan, together with co-authors Jérémy Berthomieu, Andrew Ferguson and Mohab Safey El Din (all from Sorbonne Université), studied determinantal polynomial systems. These are polynomial systems involving maximal minors of some given matrix. An important situation where these arise is the computation of the critical values of a polynomial map restricted to an algebraic set. This leads directly to a strategy for, among other problems, polynomial optimization.

Computing Gröbner bases is a classical method for solving polynomial systems in general. For practical computations, this consists of two main stages. First, a Gröbner basis is computed with respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as Sparse-FGLM, designed by Faugère and Mou, is used to find a Gröbner basis of the same system but with respect to a lexicographic ordering. The complexity of this latter step, in terms of the number of arithmetic operations in the ground field, is $O\left(m{D}^2\right)$, where $D$ is the degree of the ideal generated by the input and $m$ is the number of non-trivial columns of a certain $D\times D$ matrix.

While asymptotic estimates are known for $m$ in the case of generic polynomial systems, thus far, the complexity of Sparse-FGLM was unknown for the class of determinantal systems.

By assuming Fröberg's conjecture, a classical conjecture in commutative algebra, and thus ensuring that the Hilbert series of generic determinantal ideals have the necessary structure, the authors expand the work of Moreno-Socías by detailing the structure of the DRL staircase in the determinantal setting. Then, they study the asymptotics of the quantity $m$ by relating it to the coefficients of these Hilbert series. Consequently, they arrive at a new bound on the complexity of the Sparse-FGLM algorithm for generic determinantal systems and, in particular, for generic critical point systems.

The ideal is considered inside the polynomial ring $𝕂[x_1,\cdots ,x_n]$, where $𝕂$ is some infinite field, generated by $p$ generic polynomials of degree $d$ and the maximal minors of a $p\times (n-1)$ polynomial matrix with generic entries of degree $d-1$. Then, in this setting, for the case $d=2$ and for $n\gg p$ the paper 2 establishes an exact formula for $m$ in terms of $n$ and $p$. Moreover, for $d\ge 3$, it gives a tight asymptotic formula, as $n\to \infty$, for $m$ in terms of $n,p$ and $d$.

6.18 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 18.

6.19 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 20.

6.20 Axioms for a theory of signature bases

Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. Pierre Lairez 19 proposes an axiomatic approach to Gröbner bases with signatures with the purpose of uncoupling the theory and the algorithms, and giving general results applicable in many different settings (e.g. Gröbner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.).

6.21 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 123 and Thomas Cluzeau in 69 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 14, the author present 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.

A full article presenting this work in more details should be submitted in 2023.

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 2023. 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, 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:
  • Alexandre Goyer, Mathématiques générales pour la biologie (LSMA100A), 60h, L1, Université de Versailles Saint-Quentin-en-Yvelines, France.
  • Eric Pichon-Pharabod, Integration, Taylor expansion, Ordinary differential equations (MAA105), 64h, 2nd semester Bachelor Polytechnique, France.
  • Eric Pichon-Pharabod, Topology and multivariable calculus (MAA105), 64h, 3rd semester Bachelor Polytechnique, France.
  • Hadrien Notarantonio, Structure de données (LU2IN006), 64h, 2nd semester, Sorbonne Université, 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.
  • Frédéric Chyzak, Algorithmes efficaces en calcul formel, 9h, M2, MPRI, France. (Also responsible for the course until 2022.)
  • 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. This year, the organizers for our team were Pierre Lairez and Hadrien Notarantonio. We could have 13 talks, including some by international visiting colleagues.
• The team also runs an internal weekly meeting, where some of us could give early talks on new results. This year we had talks by Guy Fayolle, Alexandre Goyer, Alaa Ibrahim, Pierre Lairez, Raphaël Pagès, Eric Pichon-Pharabod, and Sergey Yurkevich.
• In December, the team had a 1-day visit of a group of 25 master's students from Université of Versailles Saint Quentin (M1 and M2 of the Applied Algebra master). Introductory talks about our research were given by Alin Bostan, Hadrien Brochet, Frédéric Chyzak, Alexandre Goyer, Pierre Lairez, Hadrien Notarantonio, and Eric Pichon-Pharabod.

9.1 Publications of the year

9.2 Cited publications

• 23 articleY.Y. Abdelaziz, C.C. Koutschan and J.-M.J.-M. Maillard. On Christol's conjecture.J. Phys. A53202020, 205201, 16 pagesURL: https://doi.org/10.1088/1751-8121/ab82dc
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• 24 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--133
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• 25 inproceedingsS. A.Sergei A. Abramov and M.Mark van Hoeij. Desingularization of linear difference operators with polynomial coefficients.ISSAC~'99Conference proceedingsACM1999
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• 26 articleB.Boris Adamczewski and C.Colin Faverjon. Méthode de Mahler, transcendance et relations linéaires: aspects effectifs.3022018, 557--573
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• 27 articleB.Boris Adamczewski and T.Tanguy Rivoal. Exceptional values of $E$-functions at algebraic points.5042018, 697--908
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• 28 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--211
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• 29 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 2016
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• 30 articleD.David Bailey and J. M.J. M. Borwein. High-precision numerical integration: progress and challenges.4672011, 741--754
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• 31 articleD.David Bailey, P.Peter Borwein and S.Simon Plouffe. On the rapid computation of various polylogarithmic constants.662181997, 903--913
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• 32 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--70
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• 33 articleB.Bernhard Beckermann and G.George Labahn. A uniform approach for the fast computation of matrix-type Padé approximants.1531994, 804--823
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• 34 articleJ. P.Jason P. Bell and M.Michael Coons. Transcendence tests for Mahler functions.14532017, 1061--1070
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• 35 articleO.Olivier Bernardi and M.Mireille Bousquet-Mélou. Counting colored planar maps: algebraicity results.10152011, 315--377URL: https://doi.org/10.1016/j.jctb.2011.02.003
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• 36 articleO.Olivier Bernardi, M.Mireille Bousquet-Mélou and K.Kilian Raschel. Counting quadrant walks via Tutte's invariant method.12021
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• 37 inproceedingsJ.Jérémy Berthomieu, C.Christian Eder and M.Mohab Safey El Din. Msolve: a library for solving polynomial systems.ISSAC~'21Conference proceedingsACMJuly 2021, 51--58
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• 38 inproceedingsJ.Jérémy Berthomieu and J.-C.Jean-Charles Faugère. A polynomial-division-based algorithm for computing linear recurrence relations.ISSAC~'18Conference proceedings2018, 79--86
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• 39 articleD.Daniel Bertrand and F.Frits Beukers. Équations différentielles linéaires et majorations de multiplicités.1811985, 181--192
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• 40 articleF.F. Beukers. A note on the irrationality of $\left(2\right)$ and $\left(3\right)$.1131979, 268--272
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• 41 articleF.F. Beukers. A refined version of the Siegel-Shidlovskii theorem.16312006, 369--379URL: https://doi.org/10.4007/annals.2006.163.369
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• 42 incollectionF.F. Beukers. Padé-approximations in number theory.Padé Approximation and its Applications, Amsterdam 1980888Lecture Notes in Math.Springer1981, 90--99
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• 43 articleS.Spencer Bloch, M.Matt Kerr and P.Pierre Vanhove. A Feynman integral via higher normal functions.151122015, 2329--2375
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• 44 articleS.Spencer Bloch and P.Pierre Vanhove. The elliptic dilogarithm for the sunset graph.1482015, 328--364
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• 45 bookJ.Jonathan Borwein and D.David Bailey. Mathematics by experiment.Plausible reasoning in the 21st centuryA K Peters2008, xii+377
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• 46 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 pagesCreateSpace2017
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• 47 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--102
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• 48 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--32
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• 49 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/10.1137/16M1062855
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• 50 articleA.A. Bostan and M.M. Kauers. The complete generating function for Gessel walks is algebraic.1389With an appendix by Mark van Hoeij2010, 3063--3078
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• 51 inproceedingsA.Alin Bostan, P.Pierre Lairez and B.Bruno Salvy. Creative telescoping for rational functions using the Griffiths–Dwork method.ISSAC~'13Conference proceedingsACM2013, 93--100
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• 52 articleA.Alin Bostan, P.Pierre Lairez and B.Bruno Salvy. Multiple binomial sums.80part 22017, 351--386URL: https://doi.org/10.1016/j.jsc.2016.04.002
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• 53 articleA.Alin Bostan, T.Tanguy Rivoal and B.Bruno Salvy. Explicit degree bounds for right factors of linear differential operators.5312020, 53--62
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• 54 articleM.Mireille Bousquet-Mélou and A.Arnaud Jehanne. Polynomial equations with one catalytic variable, algebraic series and map enumeration.9652006, 623--672
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• 55 articleM.Mireille Bousquet-Mélou. Square lattice walks avoiding a quadrant.1442016, 37--79
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• 56 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--142
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• 57 articleR.R. Brak and A. J.A. J. Guttmann. Algebraic approximants: a new method of series analysis.23241990, L1331--L1337
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• 58 articleN.Nils Bruin, J.Jeroen Sijsling and A.Alexandre Zotine. Numerical computation of endomorphism rings of Jacobians.2113th Algorithmic Number Theory Symposium2019, 155--171
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• 59 inproceedingsM.Manfred Buchacher, M.Manuel Kauers and G.Gleb Pogudin. Separating variables in bivariate polynomial ideals.ISSAC~'20Conference proceedingsACM2020, 54--61
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• 60 bookJ. C.J. C. Butcher. Numerical methods for ordinary differential equations.John Wiley & Sons2016, xxiii+513
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• 61 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--164
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• 62 articleS.Shaoshi Chen, M.Manuel Kauers, Z.Ziming Li and Y.Yi Zhang. Apparent singularities of D-finite systems.952019, 217--237
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• 63 incollectionS.Shaoshi Chen and M.Manuel Kauers. Order-degree curves for hypergeometric creative telescoping.ISSAC~'12Conference proceedingsACM2012, 122--129
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• 64 articleS.Shaoshi Chen and M.Manuel Kauers. Trading order for degree in creative telescoping.4782012, 968--995
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• 65 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--232
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• 66 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--68
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• 67 articleF.Frédéric Chyzak, T.Thomas Dreyfus, P.Philippe Dumas and M.Marc Mezzarobba. Computing solutions of linear Mahler equations.873142018, 2977--3021
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• 68 inproceedingsF.Frédéric Chyzak and P.Philippe Dumas. A Gröbner-basis theory for divide-and-conquer recurrences.ISSAC~'20Conference proceedingsACMJuly 2020
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• 69 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/10.1145/860854.860875
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• 70 articleE.Edgar Costa, N.Nicolas Mascot, J.Jeroen Sijsling and J.John Voight. Rigorous computation of the endomorphism ring of a Jacobian.883172019, 1303--1339
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• 71 articleJ.John Cremona. The L-functions and modular forms database project.1662016, 1541--1553
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• 72 articleS.Sławomir Cynk and D.Duco van Straten. Periods of rigid double octic Calabi-Yau threefolds.12312019, 243--258
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• 73 articleB.Bernard Deconinck and M.Mark van Hoeij. Computing Riemann matrices of algebraic curves.152-153Special issue to honor Vladimir ZakharovMay 2001, 28--46
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• 74 articleK.Karl Dilcher and K. B.Kenneth B Stolarsky. A polynomial analogue to the Stern sequence.3012007, 85--103
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• 75 articleA.Alexandru Dimca. On the de Rham cohomology of a hypersurface complement.11341991, 763--771
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• 76 articleT.Thomas Dreyfus, C.Charlotte Hardouin and J.Julien Roques. Hypertranscendance of solutions of Mahler equations.202018, 2209--2238
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• 77 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--203
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• 78 miscA.-S.Andreas-Stephan Elsenhans and J.Jörg Jahnel. Real and complex multiplication on K3 surfaces via period integration.February 2018
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• 79 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--88
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• 80 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--83
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• 81 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--122
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• 82 articleJ.-C.J.-Ch. Faugère and C.Ch. Mou. Sparse FGLM algorithms.8032017, 538--569
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• 83 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/10.1007/978-3-319-50930-3
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• 84 articleS.Stéphane Fischler and T.Tanguy Rivoal. Approximants de Padé et séries hypergéométriques équilibrées.82102003, 1369--1394
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• 85 miscS.S. Fischler and T.T. Rivoal. Effective algebraic independence of values of E-functions.Preprint arXiv2019
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• 86 bookJ.Joachim von zur Gathen and J.Jürgen Gerhard. Modern computer algebra.Cambridge University Press, Cambridge2013, xiv+795URL: https://doi.org/10.1017/CBO9781139856065
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• 87 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--427
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• 88 articleD. Y.D. Yu. Grigor'ev. Complexity of factoring and calculating the GCD of linear ordinary differential operators.1011990, 7--37
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• 89 articleA. J.A. J. Guttmann and G. S.G. S. Joyce. On a new method of series analysis in lattice statistics.591972, 81--84
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• 90 articleA. J.A. J. Guttmann. On the recurrence relation method of series analysis.871975, 1081--1088
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• 91 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-z
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• 92 articleJ. M.Johannes M. Henn. Lectures on differential equations for Feynman integrals.48152015, 153001
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• 93 articleD.Didier Henrion, J. B.Jean B. Lasserre and C.Carlo Savorgnan. Approximate volume and integration for basic semialgebraic sets.5142009, 722--743
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• 94 articleC.Charles Hermite. Sur la fonction exponentielle.771873, 18--24
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• 95 articleJ.Joris van der Hoeven. Fast evaluation of holonomic functions.21011999, 199--215
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• 96 articleJ.Joris van der Hoeven. Constructing reductions for creative telescoping.2020
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• 97 articleD. L.D. L. Hunter and G. A.G. A. Baker Jr. Methods of series analysis III. Integral approximant methods.1971979, 3808--3821
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• 98 articleA. M.A. M. Jasour, N. S.N. S. Aybat and C. M.C. M. Lagoa. Semidefinite programming for chance constrained optimization over semialgebraic sets.2532015, 1411--1440
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• 99 inproceedingsA. M.Ashkan M. Jasour, A.Andreas Hofmann and B. C.Brian C. Williams. Moment-sum-of-squares approach for fast risk estimation in uncertain environments.2018 IEEE Conference on Decision and Control (CDC)2018, 2445--2451
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• 100 articleM. A.M. A. H. Khan. High-order differential approximants.14922002, 457--468
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• 101 bookD. E.Donald E. Knuth. The art of computer programming. Vol. 2.Seminumerical algorithms, Third edition [of MR0286318]Addison-Wesley, Reading, MA1998
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• 102 articleC.Christoph Koutschan, M.Manuel Kauers and D.Doron Zeilberger. Proof of George Andrews's and David Robbins's $q$-TSPP conjecture.10862011, 2196--2199
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• 103 articleC.Christoph Koutschan and T.Thotsaporn Thanatipanonda. Advanced computer algebra for determinants.1732013, 509--523
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• 104 articleP.Pierre Lairez. Computing periods of rational integrals.853002016, 1719--1752
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• 105 inproceedingsP.Pierre Lairez, M.Marc Mezzarobba and M.Mohab Safey El Din. Computing the volume of compact semi-algebraic sets.ISSAC~'19Conference proceedingsACM2019, 259--266
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• 106 articleP.Pierre Lairez and E. C.Emre Can Sertöz. A numerical transcendental method in algebraic geometry: computation of Picard groups and related invariants.342019, 559--584
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• 107 unpublishedJ. B.Jean Bernard Lasserre. Connecting optimization with spectral analysis of tri-diagonal matrices.March 2020
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• 108 unpublishedJ. B.Jean Bernard Lasserre, V.Victor Magron, S.Swann Marx and O.Olivier Zahm. Minimizing rational functions: a hierarchy of approximations via pushforward measures.December 2020
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• 109 bookJ. B.Jean Bernard Lasserre. Moments, positive polynomials and their applications.1Series on Optimization and its ApplicationsImperial College PressOctober 2009, xxii+361
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• 110 articleJ. B.Jean Bernard Lasserre. Volume of sublevel sets of homogeneous polynomials.322019, 372--389
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• 111 articleL. F.Laura Felicia Matusevich. Weyl closure of hypergeometric systems.602June 2009, 147--158
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• 112 inproceedingsM.Marc Mezzarobba. NumGfun: a package for numerical and analytic computation and D-finite functions.ISSAC~'10Conference proceedingsACM2010, 139--146
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• 113 miscM.Marc Mezzarobba. Rigorous multiple-precision evaluation of D-finite functions in Sagemath.July 2016
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• 114 articleP.Pascal Molin and C.Christian Neurohr. Computing period matrices and the Abel-Jacobi map of superelliptic curves.883162019, 847--888
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• 115 inproceedingsS.S. Naldi and V.V. Neiger. A divide-and-conquer algorithm for computing Gröbner bases of syzygies in finite dimension.ISSAC~'20Conference proceedings2020, 380--387
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• 116 thesisC.Christian Neurohr. Efficient integration on Riemann surfaces and applications.Ph.D. ThesisUniversität Oldenburg2018
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• 117 articleT.Toshinori Oaku. Algorithms for $b$-functions, restrictions, and algebraic local cohomology groups of $D$-modules.1911997, 61--105
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• 118 articleT.Toshinori Oaku and N.Nobuki Takayama. Algorithms for $D$-modules: restriction, tensor product, localization, and local cohomology groups.1562-32001, 267--308
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• 119 articleM.Marko Petkovšek. Hypergeometric solutions of linear recurrences with polynomial coefficients.141992, 243--264
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• 120 bookG.G. Pólya. Mathematics and plausible reasoning.Induction and analogy in mathematicsPrinceton U. Press1954, xvi+280
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• 121 bookG.G. Pólya. How to solve it.A new aspect of mathematical methodPrinceton U. Press1945, xxviii+253
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