Name: Computer Algebra for Functional Equations in Combinatorics and Physics
Start: 2023-11-27T09:00:00+01:00
End: 2023-12-11T18:00:00+01:00
Location: Institut Henri Poincaré

lundi 27 novembre 2023

09:30 Creative Telescoping by Shaoshi Chen, Manuel Kauers & Christoph Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
Creative Telescoping by Shaoshi Chen, Manuel Kauers & Christoph Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
09:30 - 12:00
Room: Amphithéâtre Hermite / Darboux

15:00 A generating function method for the determination of differentially algebraic integer sequences modulo prime powers by Christian Krattenthaler. 15:00-17:00. IHP, Amphitheater Hermite
A generating function method for the determination of differentially algebraic integer sequences modulo prime powers by Christian Krattenthaler. 15:00-17:00. IHP, Amphitheater Hermite
15:00 - 17:00
Room: Amphithéâtre Hermite / Darboux
mardi 28 novembre 2023

09:30 Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
09:30 - 12:00
Room: Amphithéâtre Hermite / Darboux

15:00 A generating function method for the determination of differentially algebraic integer sequences modulo prime powers by C. Krattenthaler. 15:00-17:00. IHP, Amphitheater Hermite
A generating function method for the determination of differentially algebraic integer sequences modulo prime powers by C. Krattenthaler. 15:00-17:00. IHP, Amphitheater Hermite
15:00 - 17:00
Room: Amphithéâtre Hermite / Darboux
mercredi 29 novembre 2023

09:30 Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
09:30 - 12:00
Room: Amphithéâtre Hermite / Darboux

15:00 General audience presentation The Zeta Team by Wadim Zudilin, Radboud University. 15:00-16:00. IHP, Amphitheater Hermite
General audience presentation The Zeta Team by Wadim Zudilin, Radboud University. 15:00-16:00. IHP, Amphitheater Hermite
15:00 - 16:00
Room: Amphithéâtre Hermite / Darboux Abstract. An innocent story of finding recursions for the sums of powers of binomial coefficients transformed into an intrigue involving creative telescoping and irrationality. The goalscorers performed as a dream team.

16:00 General audience presentation. On Fermat’s penultimate theorem by Xavier Caruso, IMB, Bordeaux. 16:00-17:00. IHP
General audience presentation. On Fermat’s penultimate theorem by Xavier Caruso, IMB, Bordeaux. 16:00-17:00. IHP
16:00 - 17:00
Room: Amphithéâtre Hermite / Darboux Abstract. While reading books, Fermat often added notes in the margin for himself. The most famous one is certainly what is called nowadays Fermat's Last Theorem which stipulates that the equation xn+yn=zn has no nontrivial solution as soon as n>2. More than three centuries of effort were needed to eventually write down a proof of this result. In this presentation, I will discuss another margin-theorem by Fermat. It is less known but still quite impressive as it took almost two centuries to the community to obtain a full proof, opening as a byproduct new topics of research.
jeudi 30 novembre 2023

09:30 Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
09:30 - 12:00
Room: Amphithéâtre Hermite / Darboux

15:00 Special session of the Differential Seminar. Values of E-functions are not Liouville numbers by Tanguy Rivoal, Institut Fourier. 15:00-16:00. IHP, Amphitheater Hermite
Special session of the Differential Seminar. Values of E-functions are not Liouville numbers by Tanguy Rivoal, Institut Fourier. 15:00-16:00. IHP, Amphitheater Hermite
15:00 - 16:00
Room: Amphithéâtre Hermite / Darboux Abstract. A Liouville number is an irrational real number with an infinite irrationality exponent. Almost all real numbers are not Liouville numbers but it can be difficult in practice to prove that a given real number is not a Liouville one. In this talk, I will explain how a combination of results due to André, Beukers and Shidlovskii enables to prove that the real values of E-functions at algebraic points are not Liouville numbers. This is a joint work with Stéphane Fischler (Université Paris Saclay).

16:00 Special session of the Differential Seminar. Rounding error analysis of linear recurrences using generating series by Marc Mezzarobba, LIX, Palaiseau. 16:00-17:00. IHP, Amphitheater Hermite
Special session of the Differential Seminar. Rounding error analysis of linear recurrences using generating series by Marc Mezzarobba, LIX, Palaiseau. 16:00-17:00. IHP, Amphitheater Hermite
16:00 - 17:00
Room: Amphithéâtre Hermite / Darboux Abstract. In the computation of linearly recurrent sequences, round-off errors arising from each step tend to “cancel out” rather than just accumulate. This phenomenon is crucial to consider when aiming to establish realistic error bounds. That requires a close examination of how a given round-off error propagates through the remaining iterations of the recurrence. Doing so using traditional “sequence” notations results in intricate calculations involving nested sums. However, in other contexts, such as enumerative combinatorics, the use of generating series to encode sequences simplifies these calculations while simultaneously granting access to powerful new analytic tools. In this talk, I will show how the idea of using generating series can be adapted to the error analysis of numerical algorithms based on linear recurrence relations.
vendredi 1 décembre 2023

09:30 Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
Creative Telescoping by S. Chen, M. Kauers & C. Koutschan. 9:30-12:00. IHP, Amphitheater Hermite
09:30 - 12:00
Room: Amphithéâtre Hermite / Darboux

15:00 Special session of the Differential Seminar. Computation of sums and integrals by reduction-based creative telescoping by Bruno Salvy, Inria, ENS Lyon. 15:00-16:00. IHP, Amphitheater Hermite
Special session of the Differential Seminar. Computation of sums and integrals by reduction-based creative telescoping by Bruno Salvy, Inria, ENS Lyon. 15:00-16:00. IHP, Amphitheater Hermite
15:00 - 16:00
Room: Amphithéâtre Hermite / Darboux Abstract. Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums or integrals by synthesizing summands or integrands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums or integrals of D-finite functions as well as the associated certificates in a compact form. In the integral case, the algorithm relies on a generalization of the Hermite reduction in symbolic integration. In the sum case, the algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the Abramov-Petkovsek reduction. We present a Maple implementation with good timings on a variety of examples.

16:00 Special session of the Differential Seminar. Combinatorics and Transcendence: Applications of Inhomogeneous order 1 iterative functional equations by Marni Mishna, Simon Fraser University. 16:00-17:00. IHP, Amphitheater Hermite
Special session of the Differential Seminar. Combinatorics and Transcendence: Applications of Inhomogeneous order 1 iterative functional equations by Marni Mishna, Simon Fraser University. 16:00-17:00. IHP, Amphitheater Hermite
16:00 - 17:00
Room: Amphithéâtre Hermite / Darboux The problem of understanding the structure of transcendental objects has fascinated mathematicians for well over a century. Combinatorics provides an intuitive framework to study power series. A combinatorial family is associated to a power series in $\mathbb{Q}[[t]]$ via its enumerative generating function wherein the number of objects of size $n$ is the coefficient of $t^n$. Twentieth century combinatorics and theoretical computer science provided characterizations of classes with rational and algebraic generating functions. Finding natural extensions of these correspondences has been a motivating goal of enumerative combinatorics for several decades. This talk will focus on differentially transcendental functions. In particular, I will present recent work completed with Lucia Di Vizio and Gwladys Fernandes which characterizes solutions $f(t)$ of order 1 iterative equations of the form $f(R(t))=a(t)f(t)+b(t)$ where $R,a$, and $b$ are rational functions. These appear in the study of complete trees, walks on self-similar graphs (eg. the Sierpinski graph), and pattern avoiding permutations. The proof strategy is inspired by the Galois theory of functional equations, and relies on the property of the dynamics of $R(t)$, Liouville-Rosenlicht's theorem and Ax' theorem. This program and has led to progress in identifying the differential transcendence of combinatorial generating functions arising in the literature, and indeed generally.
samedi 2 décembre 2023
dimanche 3 décembre 2023

lundi 11 décembre 2023

09:45 Opening remarks for Topical day: Elimination for Functional Equations. IHP, Amphitheater Darboux
Opening remarks for Topical day: Elimination for Functional Equations. IHP, Amphitheater Darboux
09:45 - 10:00
Room: Amphithéâtre Hermite / Darboux

10:00 Differential Equation Invariance Axiomatization by André Platzer
Differential Equation Invariance Axiomatization by André Platzer
10:00 - 10:45
Room: Amphithéâtre Hermite / Darboux

10:45 Coffee break
Coffee break
10:45 - 11:15
Room: Amphithéâtre Hermite / Darboux

11:15 Differential elimination ideals and spectral curves by Sonia Rueda
Differential elimination ideals and spectral curves by Sonia Rueda
11:15 - 12:00
Room: Amphithéâtre Hermite / Darboux Abstract. Commuting pairs of ordinary differential operators have been related to plane algebraic curves at least since the work of Burchnall and Chaundy a century ago. This talk is devoted to the revision of some classical results, using now a differential algebra framework and, oriented to the development of algorithms based in the computation of differential resultants. The new concept of Burchnall-Chaundy ideal of a commuting pair will be presented, as the ideal of all constant coefficient bivariate polynomials satisfied by the pair. This prime ideal will be proved equal to the radical of a differential elimination ideal and the defining ideal of a plane algebraic curve, the spectral curve of a commuting pair. We are motivated by the development of a Picard-Vessiot theory for spectral problems, in the case of algebro-geometric ordinary differential operator, which are intrinsically linked to the study of integrable hierarchies.

12:00 Lunch break
Lunch break
12:00 - 14:00
Room: Amphithéâtre Hermite / Darboux

14:00 Geometry-driven algorithms for combinatorial functional equations by Hadrien Notarantonio
Geometry-driven algorithms for combinatorial functional equations by Hadrien Notarantonio
14:00 - 14:30
Room: Amphithéâtre Hermite / Darboux Abstract. Enumerative combinatorics contains a vast landscape of problems that could hardly be solved without the consideration of special functional equations called “Discrete Differential Equations”. Among these problems, the enumeration of walks, planar maps carrying hard particles, etc. These functional equations relate formal power series in n variables with specializations of them to some of the variables (the specializations being generating functions related to the enumeration of interest). When the involved variables are “nested”, a celebrated result by Popescu (1986) implies algebraicity of the solutions. In 2006, Bousquet-Mélou and Jehanne provided an elementary proof of algebraicity of the solutions in the case $n = 2$. Their proof yields an algorithm, and it has been the state-of-the-art in enumerative combinatorics for solving these equations since then. In this talk, I will present a recent approach, based on the intensive use of effective algebraic geometry, in order to solve more efficiently such equations in the case $n = 2$. Also, I will introduce and discuss recent advances in the case of systems of such equations. The talk is based on joint works with Alin Bostan, Mohab Safey El Din and Sergey Yurkevich.

14:30 Examples of use of functional equations obtained from the elimination theory in nonlinear models by Nathalie Verdière
Examples of use of functional equations obtained from the elimination theory in nonlinear models by Nathalie Verdière
14:30 - 15:15
Room: Amphithéâtre Hermite / Darboux Abstract. This talk addresses the use of functional equations obtained from computer algebra in the structural properties study of nonlinear dynamical models. For several years, they were exploited in the context of identifiability and diagnosticability studies. A final application was recently developed and concerns an a priori study for the reconstruction of some variables of interest in nonlinear complex networks with an application on a C. elegans chemotaxis network.

15:15 Coffee break
Coffee break
15:15 - 15:45
Room: Amphithéâtre Hermite / Darboux

15:45 On some functional equations for maps by Alexandros Singh
On some functional equations for maps by Alexandros Singh
15:45 - 16:15
Room: Amphithéâtre Hermite / Darboux Abstract. The goal of this exploratory talk is to present various functional equations related to the enumeration of maps and the study of various combinatorial parameters thereof. In particular, we'll concern ourselves with questions such as: how are these equations related and can they be inter-derived by combinatorial or differentially-algebraic methods? This presentation draws from material from ongoing joint work with Olivier Bodini and Konstantinos Tsagkaris.

16:15 Solving differential elimination problems with Thomas decomposition by Daniel Robertz
Solving differential elimination problems with Thomas decomposition by Daniel Robertz
16:15 - 17:00
Room: Amphithéâtre Hermite / Darboux Abstract. This talk gives an introduction to the Thomas decomposition method for systems of nonlinear partial differential equations. A Thomas decomposition is a finite family of so-called simple differential systems, each of which is formally integrable, and such that the solution set of the given PDE system is the disjoint union of the solution sets of the simple systems. This versatile technique allows solving differential elimination problems, for example, for the analysis of certain structural properties of nonlinear control systems. The Thomas decomposition method has been implemented in Maple.

17:00 Closing remarks
Closing remarks
17:00 - 17:15
Room: Amphithéâtre Hermite / Darboux