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  52, 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  93. 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  143. 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  128, 127. 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 57109, 110. 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  64 or differential approximation  107, 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  96, 97, 104, 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  40; 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 108, 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  37. 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 $\pi$  38 (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 Awards

• Claudia Fevola was awarded the Otto Hahn Medal for the year 2024 by the Max Planck Society

6.1 Continued fractions, orthogonal polynomials and Dirichlet series

Using an experimental mathematics approach, Alin Bostan together with Frédéric Chapoton (CNRS, IRMA Strasbourg) obtained in 3 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. The article was published in 2024.

6.2 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 5 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. The article was published in 2024.

6.3 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 2, 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. The article was published in 2024.

6.4 Differential transcendence of Bell numbers and relatives: a Galois theoretic approach

In 2003, Klazar proved that the ordinary generating function of the sequence of Bell numbers is differentially transcendental over the field $C\left(\right\{t\left\}\right)$ of meromorphic functions at 0. In 4, Alin Bostan, Lucia Di Vizio, Kilian Raschel showed that Klazar's result is an instance of a general phenomenon that can be proven in a compact way using difference Galois theory. They presented the main principles of this theory in order to prove a general result about differential transcendence over $C\left(\right\{t\left\}\right)$ that they applied to many other (infinite classes of) examples of generating functions, including as very special cases the ones considered by Klazar. Most of the examples belong to Sheffer's class, well studied notably in umbral calculus. They all bring concrete evidence in support to the Pak–Yeliussizov conjecture, according to which a sequence whose both ordinary and exponential generating functions satisfy nonlinear differential equations with polynomial coefficients necessarily satisfies a linear recurrence with polynomial coefficients.

6.5 An arithmetic characterization of some algebraic functions and a new proof of an algebraicity prediction by Golyshev

In 1, Alin Bostan provided a new arithmetic characterization for the sequence of coefficients of algebraic power series $f\left(t\right)$ having the property that the differential equation $y^{\text{'}}\left(t\right)=f\left(t\right)y\left(t\right)$ has algebraic solutions only. This extends a recent result by Delaygue and Rivoal, and also provides a new and shorter proof of an algebraicity result predicted by Golyshev.

6.6 Isoperimetric ratios of toroidal Dupin cyclides

The combination of recent results due to Yu and Chen [Proc. AMS 150(4), 2020, 174–1765] and to Bostan and Yurkevich [Proc. AMS 150(5), 2022, 2131–2136] shows that the 3-D Euclidean shape of the square Clifford torus is uniquely determined by its isoperimetric ratio. This solves part of the still open uniqueness problem of the Canham model for biomembranes. In 20, Alin Bostan, together with Thomas Yu and Sergey Yurkevich, investigated the generalization of the aforementioned result to the case of a rectangular Clifford torus. Like in the square case, they found closed form formulas in terms of hypergeometric functions for the isoperimetric ratio of its stereographic projection to ${ℝ}^3$ and showed that the corresponding function is strictly increasing. But unlike in the square case, they showed that the isoperimetric ratio does not uniquely determine the Euclidean shape of a rectangular Clifford torus.

6.7 Positivity certificates for linear recurrences

In 16, 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. The article was published in 2024.

6.8 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 13 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.). This is a work of Pierre Lairez, together with Christian Eder, Rafael Mohr and Mohab Safey El Din 84.

6.9 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 6 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. The article was published in 2024.

6.10 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 8 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. The article was published in 2024.

6.11 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 7, 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. The article was published in 2024.

6.12 Effective homology and periods of complex projective hypersurfaces

In 14, 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. The article was published in 2024.

6.13 A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over the projective line

In 30, Eric Pichon-Pharabod provided an algorithm to compute an effective description of the homology and periods of elliptic surfaces extending the methods of 14. This is a novel algorithm allowing to compute the periods of previously inaccessible surfaces. In particular, for quartic K3 surfaces which have an elliptic fibration, this greatly decreases the computation time. This algorithm them allows to recover both heuristic and exact invariants of the elliptic surface, such as the integral monodromy representation and the Mordell-Weil lattice. The algorithm comes with a SageMath implementation. The article was published in 2024.

6.14 Conway's cosmological theorem and automata theory

John Conway proved that every audioactive sequence (a.k.a. look-and-say) decays into a compound of 94 elements, a statement he termed the cosmological theorem. The underlying audioactive process can be modeled by a finite-state machine, mapping one sequence of integers to another. Leveraging automata theory, Pierre Lairez and Aleksandr Storozhenko propose a new proof of Conway's theorem based on a few simple machines, using a computer to compose and minimize them 28.

6.15 Validated numerics for algebraic path tracking

Using validated numerical methods, interval arithmetic and Taylor models, Alexandre Guillemot and Pierre Lairez 15 propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. They provide a Rust implementation which shows tremendous improvement over existing software for certified path tracking.

6.16 Wronski pairs of honeycomb curves

In 21 Laura Casabella (MPI-MiS, Leipzig), Michael Joswig (TU Berlin) and Rafael Mohr studied certain generic systems of real polynomial equations associated with triangulations of convex polytopes and investigated their number of real solutions. The main focus was set on pairs of plane algebraic curves which form a so-called Wronski system. The computational tasks arising in the analysis of such Wronski pairs lead the authors to the frontiers of current computer algebra algorithms and their implementations, both via Gröbner bases and numerical algebraic geometry.

6.17 Differential equations satisfied by generating functions of 5-, 6-, and 7-regular labelled graphs: a reduction-based approach

By a classic result of Gessel, the exponential generating functions for $k$-regular graphs are D-finite. Using Gröbner bases in Weyl algebras, Frédéric Chyzak and Marni Mishna (Simon Fraser University) computed the linear differential equations satisfied by the generating function for 5-, 6-, and 7- regular graphs 23. Their method is sufficiently robust to consider variants such as graphs with multiple edges, loops, and graphs whose degrees are limited to fixed sets of values.

6.18 First-order factors of linear Mahler operators

In 22, Frédéric Chyzak and Philippe Dumas, together with Thomas Dreyfus (Université de Bourgogne) and Marc Mezzarobba (LIX), developed and compared two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators ${\ell }_r{M}^r+\cdots +{\ell }_{1}M+{\ell }_0$ where ${\ell }_0,\cdots ,{\ell }_r$ are polynomials in $x$ and $Mx=x^{b}M$ for some integer $b\ge 2$. In other words, they gave algorithms for finding all formal infinite product solutions of linear functional equations ${\ell }_r\left(x\right)f\left(x^{b^r}\right)+\cdots +{\ell }_1\left(x\right)f\left(x^b\right)+{\ell }_0\left(x\right)f\left(x\right)=0$.

The first of their algorithms is adapted from Petkovšek's classical algorithm for the analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite–Padé approximants to detect those linear combinations of the solutions that correspond to first-order factors.

In their article, which was submitted and revised in 2024, they presented implementations of both algorithms and discussed their use in combination with criteria from the literature to prove the differential transcendence of power series solutions of Mahler equations.

6.19 Asymptotics and time-scaling of stop-and-go waves in car-following models

Waves, known as stop-and-go waves or phantom jams, can appear spontaneously in dense traffic. This causes a situation where drivers are faced with consecutive phases of acceleration and braking. Although waves are well understood in the setting of macroscopic models, there were no such results for the car-following model. Starting from the linearization of this model, G. Fayolle and J.M. Lagouttes (Inria-Paris) gave an asymptotic approximation of the speed and shape of these waves 25. Their method relies on a sharp application of the well-known saddle-point method, in order to describe the trajectory of a vehicle caught in such a wave.

6.20 Thermodynamical limits for models of car-sharing systems: the Autolib' example

Ch. Fricker (Inria-Paris) and G. Fayolle analyzed in 24 various mean-field equations obtained for models involving a large station-based car-sharing system in France called Autolib'. Their focus is mainly on a version without capacity constraints, where users reserve a parking space when they take a car. The model is carried out in thermodynamical limit, that is when the number $N$ of stations and the fleet size $M_N$ tend to infinity with $U={lim}_{N\to \infty }{M}_N/N$. This limit is described by Kolmogorov's equations of a two-dimensional time-inhomogeneous Markov process depicting the numbers of reservations and cars at a station. It satisfies a non-linear differential system having a unique solution, which converges, as $t\to \infty$, exponentially fast towards an equilibrium point, which corresponds to the stationary distribution of a two-queue tandem (reservations, cars), that is always ergodic. The intensity factor of each queue has an explicit form obtained from an intrinsic mass conservation relationship. Two related models with capacity constraints were also presented: the simplest one with no reservation leads to a one-dimensional problem; the second one corresponds to our first model with finite total capacity $K$.

6.21 A Markovian analysis of IEEE 802.11 broadcast transmission networks with buffering and back-off stages

G. Fayolle and P. Mühlethaler (Inria-Paris) analyzed 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. The key point of the 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, stability (ergodicity) conditions are obtained and interpreted in terms of maximum throughput. The article  9 appeared in 2024.

6.22 Landau singularities revisited: computational algebraic geometry for Feynman integrals

Claudia Fevola together with Sebastian Mizera (Institute for Advanced Studies) and Simon Telen (Max Planck Intitute for Mathematics in the Sciences) in 10 reformulated the analysis of singularities of Feynman integrals in a way that can be practically applied to perturbative computations in the Standard Model in dimensional regularization. After highlighting issues in the textbook treatment of Landau singularities, they developed an algorithm for classifying and computing them using techniques from computational algebraic geometry. They introduced an algebraic variety called the principal Landau determinant, which captures the singularities even in the presence of massless particles or UV/IR divergences and illustrated this for 114 example diagrams, including a cutting-edge 2-loop 5-point non-planar QCD process with multiple mass scales.

6.23 Principal Landau determinants

Claudia Fevola together with Sebastian Mizera (Institute for Advanced Studies) and Simon Telen (Max Planck Intitute for Mathematics in the Sciences) in 11 reformulated the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. They contributed new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, the authors defined the principal Landau determinant of a Feynman diagram. They illustrated with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus, and adapted the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities were detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. Fevola, Mizera, and Telen computed principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Their algorithms build on the Julia package Landau.jl and were implemented in the new open-source package PLD.jl.

6.24 Algebraic approaches to cosmological integrals

Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools 27, Claudia Fevola together with Guilherme L. Pimentel (Scuola Normale Superiore and INFN), Anna-Laura Sattelberger (Max Planck Institute for Mathematics in the Sciences), and Tom Westerdijk (Scuola Normale Superiore and INFN) shed light on the differential and difference equations satisfied by these integrals. Moreover, they studied a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.

6.25 Euler discriminant of complements of hyperplanes

The Euler discriminant of a family of very affine varieties is defined as the locus where the Euler characteristic drops. Claudia Fevola together with Saiei J. Matsubara-Heo (Kumamoto University) studied the Euler discriminant of families of complements of hyperplanes. They proved 26 that the Euler discriminant is a hypersurface in the space of coefficients, and provide its defining equation in two cases: (1) when the coefficients are generic, and (2) when they are constrained to a proper subspace. In the generic case, the authors showed that the multiplicities of the components can be recovered combinatorially. This analysis also recovers the singularities of an Euler integral. In the appendix, a relation to cosmological correlators is discussed.

6.26 Solving the $p$-Riccati equations and applications to the factorisation of differential operators

The solutions of the equation $f^{(p-1)}+f^p=h^p$ in the unknown function $f$ over an algebraic function field of characteristic $p$ are very closely linked to the structure and factorisations of linear differential operators with coefficients in function fields of characteristic $p$. However, while being able to solve this equation over general algebraic function fields is necessary even for operators with rational coefficients, no general resolution method has been developed. Raphaël Pagès presented 29 an algorithm for testing the existence of solutions in polynomial time in the “size” of $h$ and an algorithm based on the computation of Riemann–Roch spaces and the selection of elements in the divisor class group, for computing solutions of size polynomial in the “size” of $h$ in polynomial time in the size of $h$ and linear in the characteristic $p$. He also discussed its applications to the factorisation of linear differential operators in positive characteristic $p$.

6.27 Geometry-driven algorithms for the efficient solving of combinatorial functional equations

Hadrien Notarantonio's thesis lies at the intersection of computer algebra and enumerative combinatorics. From a given enumeration problem, a well known habit in enumerative combinatorics consists in turning the enumeration problem into a functional equation involving the generating function associated to the enumeration. In some sophisticated enumeration problems also appear systems of functional equations. The goal is to study these functional equations in order to deduce informations on the initial enumeration problem. A celebrated result by Bousquet-Mélou and Jehanne in 2006 states that when these functional equations possess a particular shape, their solutions are unique and are annihilated by nonzero polynomials. They also provide a first algorithm for computing such polynomials.

In the thesis that he completed this year 17, Hadrien Notarantonio relied on classical results from computer algebra in order to analyze and compare (i.e., with quantitative estimates) the state-of-the-art algorithms. Based on these theoretical analyses, he then designed efficient geometry-driven algorithms for computing these annihilating polynomials. The new algorithms have then been implemented in a software: this made it possible to solve previously out-of-reach problems via automated approaches. Last but not least, he extended the state-of-the-art to the case of systems of functional equations, with some views in the direction of Artin’s approximation theory.

6.28 Homology and periods of algebraic varieties

In his thesis 19, which he completed this year, Eric Pichon-Pharabod focused on the development of semi-numerical algorithms for the evaluation of high precision numerical approximations of periods of algebraic varieties from equations that define these varieties. The periods of a smooth projective algebraic variety determine the de Rham isomorphism between the de Rham cohomology and the singular homology, and define a continuous fine invariant of the variety in relation to its Hodge structure. In many cases, Torelli theorems show that periods fully characterize the class of variety isomorphism — this is for example the case for surfaces K3. In particular, the knowledge of sufficiently precise numerical approximations (around 100 significant digits) of these periods allows to find fine geometric invariants of the variety, as for example the Picard rank and the body of endomorphisms for K3 surfaces.

Eric Pichon-Pharabod developed a method to calculate such numerical approximations from the equations defining the variety. This method relies on the theory of Picard-Lefschetz to reconstruct a description of integration cycles that is sufficiently explicit to be able to evaluate these integrals. He implemented these methods for the case of hypersurfaces, elliptic surfaces and double overlays of P2 branched along a curve, as well as some varieties of Calabi–Yau of dimension 3. He used these methods to explore the phenomenology of some open problems in mirror symmetry and arithmetic geometry.

6.29 Factorization of linear differential operators in positive characteristic

The study of linear differential operators is an important part of the algebraic study of differential equations. The rings of operators linear differentials share many properties with the rings of polynomials, but the non-commutative character of multiplication makes the design of factorization algorithms more complicated. The object of Raphaël Pagès's thesis 18 was the development of an algorithm for calculating a right-hand irreducible factor of a given linear differential operator whose coefficients are elements of an algebraic function field of characteristic $p$. The situation differs greatly from the similar problem in characteristic 0 because the algebraic function fields of positive characteristic are finite-dimensional over their field of constants. From this results an Azumaya algebra structure that provides additional tools to tackle the factorization problem.

A first step is the calculation of the $p$-curvature, a classical invariant of first importance of differential operators in characteristic $p$. Given differential operator $L$ in characteristic 0 and a given integer $N\in \mathbb{N}$, the first significant result of the thesis is a computational algorithm for computing all the characteristic polynomials of the $p$-curvatures of the reductions of $L$ modulo $p$, for all prime numbers $p\ge N$.

The second part of the thesis is devoted to factorization in itself. Raphaël Pagès used the Azumaya algebra structure to show that the search for irreducible right-hand factors reduces to solving the $p$-Riccati equation

for its solutions in $K\left[a\right]$, where $a$ is some algebraic function on $K$. This observation allowed him to develop two important algorithms. The first is an application of the global-local principle leading to an irreducibility test of polynomial complexity for differential operators. The second is a solving algorithm for the $p$-Riccati equation using several tools from the algebraic geometry of curves, including Riemann–Roch spaces and Picard groups. He performed a thorough complexity analysis of this algorithm and showed that the $p$-Riccati equation always admits a solution whose size is comparable to that of the parameter $a$. This algorithm makes it in particular possible to factor central operators (a case that has often been left out by the past) and to decrease the size of irreducible right-hand factors of differential linear operators by a factor $p$ in comparison to previous works. Raphaël Pagès finally deduced from this a complete factorization algorithm for linear differential operators in positive characteristic.

7.1 International initiatives

7.2 European initiatives

7.3 National initiatives

• De rerum natura. This project, set up by the team, was accepted in 2019 and has been funded until June 2024. It gathered over 20 experts from four fields: computer algebra; the Galois theories of linear functional equations; number theory; combinatorics and probability. Our goal has been 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 and terminated in November 2024. Its objective was 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 aimed 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.)

8.1 Promoting scientific activities

8.2 Teaching - Supervision - Juries

• Bachelor:
  • Ricardo Buring, Mécanismes de la programmation orientée-objet (INF371), 40h, Bachelor Polytechnique, France.
  • Ricardo Buring, Object-oriented Programming in C++ (CSE201), 28h, Bachelor Polytechnique, France.
  • Alexandre Guillemot, Linear Algebra (MAA101), 64h, 1st semester, Bachelor Polytechnique, France.
  • Eric Pichon-Pharabod, Introduction to Analysis (MAA102), 64h, 1st semester Bachelor Polytechnique, France.
  • Claudia Fevola, Machine Learning (MAA101), 28h, 1st semester, Bachelor Polytechnique, France.
• Master:
  • Alin Bostan, Algorithmes efficaces en calcul formel, 18h, 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.
  • Alexandre Guillemot, 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. Beyond the direct interest of the team, the seminar is open to remote attendance in hybrid mode. In 2024, it received 11 speakers, including 7 from abroad. Five speakers prolonged their stay to up to a week, in the framework of a collaboration with members of the team.

9.1 Publications of the year

9.2 Cited publications

• 31 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
• 32 inproceedingsS. A.Sergei A. Abramov and M.Mark van Hoeij. Desingularization of linear difference operators with polynomial coefficients.ISSAC~'99Conference proceedingsACM1999DOIback to text
• 33 articleB.Boris Adamczewski and C.Colin Faverjon. Méthode de Mahler, transcendance et relations linéaires: aspects effectifs.3022018, 557--573back to text
• 34 articleB.Boris Adamczewski and T.Tanguy Rivoal. Exceptional values of $E$-functions at algebraic points.5042018, 697--908back to textback to text
• 35 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
• 36 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
• 37 articleD.David Bailey and J. M.J. M. Borwein. High-precision numerical integration: progress and challenges.4672011, 741--754DOIback to textback to text
• 38 articleD.David Bailey, P.Peter Borwein and S.Simon Plouffe. On the rapid computation of various polylogarithmic constants.662181997, 903--913DOIback to text
• 39 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
• 40 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
• 41 articleJ. P.Jason P. Bell and M.Michael Coons. Transcendence tests for Mahler functions.14532017, 1061--1070back to text
• 42 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
• 43 articleO.Olivier Bernardi, M.Mireille Bousquet-Mélou and K.Kilian Raschel. Counting quadrant walks via Tutte's invariant method.12021DOIback to text
• 44 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
• 45 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
• 46 articleD.Daniel Bertrand and F.Frits Beukers. Équations différentielles linéaires et majorations de multiplicités.1811985, 181--192back to text
• 47 articleF.F. Beukers. A note on the irrationality of $\left(2\right)$ and $\left(3\right)$.1131979, 268--272back to text
• 48 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
• 49 incollectionF.F. Beukers. Padé-approximations in number theory.Padé Approximation and its Applications, Amsterdam 1980888Lecture Notes in Math.Springer1981, 90--99back to text
• 50 articleS.Spencer Bloch, M.Matt Kerr and P.Pierre Vanhove. A Feynman integral via higher normal functions.151122015, 2329--2375DOIback to textback to text
• 51 articleS.Spencer Bloch and P.Pierre Vanhove. The elliptic dilogarithm for the sunset graph.1482015, 328--364DOIback to text
• 52 bookJ.Jonathan Borwein and D.David Bailey. Mathematics by experiment.Plausible reasoning in the 21st centuryA K Peters2008, xii+377back to text
• 53 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
• 54 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
• 55 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
• 56 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
• 57 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
• 58 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
• 59 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
• 60 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
• 61 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
• 62 articleM.Mireille Bousquet-Mélou. Square lattice walks avoiding a quadrant.1442016, 37--79back to text
• 63 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
• 64 articleR.R. Brak and A. J.A. J. Guttmann. Algebraic approximants: a new method of series analysis.23241990, L1331--L1337DOIback to text
• 65 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
• 66 inproceedingsM.Manfred Buchacher, M.Manuel Kauers and G.Gleb Pogudin. Separating variables in bivariate polynomial ideals.ISSAC~'20Conference proceedingsACM2020, 54--61DOIback to text
• 67 bookJ. C.J. C. Butcher. Numerical methods for ordinary differential equations.John Wiley & Sons2016, xxiii+513DOIback to text
• 68 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
• 69 articleS.Shaoshi Chen, M.Manuel Kauers, Z.Ziming Li and Y.Yi Zhang. Apparent singularities of D-finite systems.952019, 217--237DOIback to text
• 70 incollectionS.Shaoshi Chen and M.Manuel Kauers. Order-degree curves for hypergeometric creative telescoping.ISSAC~'12Conference proceedingsACM2012, 122--129back to text
• 71 articleS.Shaoshi Chen and M.Manuel Kauers. Trading order for degree in creative telescoping.4782012, 968--995DOIback to text
• 72 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
• 73 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
• 74 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
• 75 inproceedingsF.Frédéric Chyzak and P.Philippe Dumas. A Gröbner-basis theory for divide-and-conquer recurrences.ISSAC~'20Conference proceedingsACMJuly 2020DOIback to text
• 76 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
• 77 articleJ.John Cremona. The L-functions and modular forms database project.1662016, 1541--1553DOIback to text
• 78 articleS.Sławomir Cynk and D.Duco van Straten. Periods of rigid double octic Calabi-Yau threefolds.12312019, 243--258DOIback to textback to text
• 79 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
• 80 articleK.Karl Dilcher and K. B.Kenneth B Stolarsky. A polynomial analogue to the Stern sequence.3012007, 85--103back to text
• 81 articleA.Alexandru Dimca. On the de Rham cohomology of a hypersurface complement.11341991, 763--771DOIback to text
• 82 articleT.Thomas Dreyfus, C.Charlotte Hardouin and J.Julien Roques. Hypertranscendance of solutions of Mahler equations.202018, 2209--2238back to text
• 83 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
• 84 articleC.Christian Eder, P.Pierre Lairez, R.Rafael Mohr and M.Mohab Safey El Din. A Signature-based Algorithm for Computing the Nondegenerate Locus of a Polynomial System.Journal of Symbolic Computation119November 2023, 1-21HALDOIback to text
• 85 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
• 86 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
• 87 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
• 88 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
• 89 articleJ.-C.J.-Ch. Faugère and C.Ch. Mou. Sparse FGLM algorithms.8032017, 538--569DOIback to text
• 90 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
• 91 articleS.Stéphane Fischler and T.Tanguy Rivoal. Approximants de Padé et séries hypergéométriques équilibrées.82102003, 1369--1394DOIback to text
• 92 miscS.S. Fischler and T.T. Rivoal. Effective algebraic independence of values of E-functions.Preprint arXiv2019back to text
• 93 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
• 94 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
• 95 articleD. Y.D. Yu. Grigor'ev. Complexity of factoring and calculating the GCD of linear ordinary differential operators.1011990, 7--37DOIback to text
• 96 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
• 97 articleA. J.A. J. Guttmann. On the recurrence relation method of series analysis.871975, 1081--1088back to text
• 98 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
• 99 articleJ. M.Johannes M. Henn. Lectures on differential equations for Feynman integrals.48152015, 153001DOIback to text
• 100 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
• 101 articleC.Charles Hermite. Sur la fonction exponentielle.771873, 18--24back to text
• 102 articleJ.Joris van der Hoeven. Fast evaluation of holonomic functions.21011999, 199--215DOIback to text
• 103 articleJ.Joris van der Hoeven. Constructing reductions for creative telescoping.2020DOIback to text
• 104 articleD. L.D. L. Hunter and G. A.G. A. Baker Jr. Methods of series analysis III. Integral approximant methods.1971979, 3808--3821DOIback to text
• 105 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--1440DOIback to text
• 106 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--2451DOIback to textback to text
• 107 articleM. A.M. A. H. Khan. High-order differential approximants.14922002, 457--468DOIback to text
• 108 bookD. E.Donald E. Knuth. The art of computer programming. Vol. 2.Seminumerical algorithms, Third edition [of MR0286318]Addison-Wesley, Reading, MA1998back to text
• 109 articleC.Christoph Koutschan, M.Manuel Kauers and D.Doron Zeilberger. Proof of George Andrews's and David Robbins's $q$-TSPP conjecture.10862011, 2196--2199DOIback to text
• 110 articleC.Christoph Koutschan and T.Thotsaporn Thanatipanonda. Advanced computer algebra for determinants.1732013, 509--523DOIback to text
• 111 articleP.Pierre Lairez. Computing periods of rational integrals.853002016, 1719--1752DOIback to text
• 112 inproceedingsP.Pierre Lairez, M.Marc Mezzarobba and M.Mohab Safey El Din. Computing the volume of compact semi-algebraic sets.ISSAC~'19Conference proceedingsACM2019, 259--266DOIback to text
• 113 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--584DOIback to textback to text
• 114 unpublishedJ. B.Jean Bernard Lasserre. Connecting optimization with spectral analysis of tri-diagonal matrices.March 2020back to text
• 115 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 2020back to text
• 116 bookJ. B.Jean Bernard Lasserre. Moments, positive polynomials and their applications.1Series on Optimization and its ApplicationsImperial College PressOctober 2009, xxii+361DOIback to text
• 117 articleJ. B.Jean Bernard Lasserre. Volume of sublevel sets of homogeneous polynomials.322019, 372--389DOIback to text
• 118 articleL. F.Laura Felicia Matusevich. Weyl closure of hypergeometric systems.602June 2009, 147--158DOIback to text
• 119 inproceedingsM.Marc Mezzarobba. NumGfun: a package for numerical and analytic computation and D-finite functions.ISSAC~'10Conference proceedingsACM2010, 139--146DOIback to text
• 120 miscM.Marc Mezzarobba. Rigorous multiple-precision evaluation of D-finite functions in Sagemath.July 2016back to text
• 121 articleP.Pascal Molin and C.Christian Neurohr. Computing period matrices and the Abel-Jacobi map of superelliptic curves.883162019, 847--888DOIback to text
• 122 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--387back to text
• 123 thesisC.Christian Neurohr. Efficient integration on Riemann surfaces and applications.Ph.D. ThesisUniversität Oldenburg2018back to text
• 124 articleT.Toshinori Oaku. Algorithms for $b$-functions, restrictions, and algebraic local cohomology groups of $D$-modules.1911997, 61--105DOIback to text
• 125 articleT.Toshinori Oaku and N.Nobuki Takayama. Algorithms for $D$-modules: restriction, tensor product, localization, and local cohomology groups.1562-32001, 267--308DOIback to text
• 126 articleM.Marko Petkovšek. Hypergeometric solutions of linear recurrences with polynomial coefficients.141992, 243--264back to text
• 127 bookG.G. Pólya. Mathematics and plausible reasoning.Induction and analogy in mathematicsPrinceton U. Press1954, xvi+280back to text
• 128 bookG.G. Pólya. How to solve it.A new aspect of mathematical methodPrinceton U. Press1945, xxviii+253back to text
• 129 articleD.Dorin Popescu. General Néron desingularization and approximation.1041986, 85--115URL: https://doi.org/10.1017/S0027763000022698DOIback to text
• 130 articleJ.Julien Roques. On the algebraic relations between Mahler functions.37012018, 321--355back to textback to text
• 131 articleE. C.Emre Can Sertöz. Computing periods of hypersurfaces.883202019, 2987--3022DOIback to textback to text
• 132 articleA. B.A. B. Šidlovskiĭ. On transcendentality of the values of a class of entire functions satisfying linear differential equations.1051955, 35--37back to text
• 133 incollectionC. L.Carl L. Siegel. Über einige Anwendungen diophantischer Approximationen [reprint of Abhandlungen der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 1929, Nr. 1].On some applications of Diophantine approximations2Quad./Monogr.Ed. Norm., Pisa2014, 81--138back to text
• 134 inproceedingsM. F.Michael F. Singer. Algebraic solutions of $n$th order linear differential equations.Proceedings of the Queen's Number Theory Conference, 1979 (Kingston, Ont., 1979)54Queen's Papers in Pure and Appl. Math.Queen's Univ., Kingston, Ont.1980, 379--420back to text
• 135 articleL.Lucas Slot and M.Monique Laurent. Near-optimal analysis of Lasserre’s univariate measure-based bounds for multivariate polynomial optimization.2020DOIback to text
• 136 articleA. V.A. V. Smirnov and M. N.M. N. Tentyukov. Feynman integral evaluation by a sector decomposition approach (FIESTA).18052009, 735--746DOIback to text
• 137 articleV. N.V. N. Sorokin. A transcendence measure for ${}^2$.187121996, 1819--1852back to text
• 138 articleR. P.R. P. Stanley. Differentiably finite power series.121980, 175--188DOIback to text
• 139 incollectionH.Harrison Tsai. Algorithms for associated primes, Weyl closure, and local cohomology of $D$-modules.Local cohomology and its applications (Guanajuato, 1999)226Lecture Notes in Pure and Appl. Math.Dekker2002, 169--194back to text
• 140 articleH.Harrison Tsai. Weyl closure of a linear differential operator.294-5Special issue on Symbolic Computation in Algebra, Analysis, and Geometry (Berkeley, CA, 1998)2000, 747--775URL: http://dx.doi.org/10.1006/jsco.1999.0400DOIback to text
• 141 articleW. T.W. T. Tutte. Chromatic sums for rooted planar triangulations: the cases $=1$ and $=2$.251973, 426--447URL: https://doi.org/10.4153/CJM-1973-043-3DOIback to text
• 142 incollectionD.Don Zagier. The arithmetic and topology of differential equations.European Congress of MathematicsEuropean Mathematical Society, Zürich2018, 717--776back to text
• 143 articleD.Doron Zeilberger and W.Wadim Zudilin. The irrationality measure of $$ is at most 7.103205334137….942020, 407--419DOIback to textback to text