To celebrate the end of the project *De rerum natura*,
DRN joins with EFI
(*Functional Equations and Interactions*)
to organize a conference.
The meeting will gather experts of all the involved fields,
members of the project and of EFI, and invited colleagues alike:
number theory, combinatorics, computer algebra,
Galois theory of functional equations, probabilities.
The members of the project will report on the main results of the project,
colleagues of EFI and invited colleagues will present recent results on related topics.

The 1-week meeting will happen in-person in a residential conference venue in Anglet (France) on June 09–14, 2024 (detailed information at bottom of page).

Beside talks of a usual 60-minute length, long talks of 90 minutes have been selected as summaries of the contributions of the project or as surveys of neighboring topics of interest to DRN+EFI.

The program is now available. Abstracts are available below (press on the triangles on the left).

Long talks (90 minutes):

## Boris Adamczewski,

*E- and M-functions*. [slides]*Abstract:*The study of algebraic or linear relations between the values of Siegel's E-functions and Mahler's M-functions at algebraic points has been a problem studied since the late 1920s. Mysteriously enough, these two very different classes of analytic functions now give rise to twin theories. In this talk, I will review the main achievements concerning these two theories, highlighting the main issues at stake as well as recent results obtained by members of the De Rerum Natura project.## Frits Beukers,

*A sketch of Dwork's Frobenius structure*. [slides]*Abstract:*We discuss some aspects of the arithmetic of linear differential equations which are associated to algebraic geometry (Picard-Fuchs equations). In the 1960's Bernard Dwork made some fundamental discoveries in this subject in the form of $p$-adic properties of the solutions of differential equations and so-called Frobenius structures. Many of these ideas appeared in his famous paper “$p$-adic cycles”. Meanwhile the subject has continued to develop in directions of $p$-adic and crystalline cohomology. In this lecture we shall give an elementary exposition of some of Dwork's original ideas in a simplified language.## Valentin Bonzom,

*A gentle introduction to integrable hierarchies and an application in enumerative combinatorics*. [slides]*Abstract:*

An integrable hierarchy is a countably infinite set of partial differential equations for an unknown function in a countably infinite set of variables, such that the flows with respect to these variables commute with one another. In this introduction to the topic, I will make this statement concrete by describing explicitly the Toda, KdV and KP hierarchies. I will describe two classical formalisms: the Lax formulation and the $\tau$-function formulation and I will explain how to write the PDEs explicitly in both. Finally I will present an application to the enumeration of maps, which are graphs embedded in surfaces. They are known to “satisfy the KP hierarchy” and the latter can be used to extract an extremely efficient recurrence relation on triangulations of genus $g$ with $n$ triangles, as shown by Goulden and Jackson in 2008.

*References:*- For a good introduction to the topic:
*Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras*, by Date, Jimbo, Miwa. - For an impressive account of the approaches to classical integrable systems and a large number of examples:
*Introduction to Classical Integrable Systems*, by Babelon, Bernard, Talon. - For an algebraic approach to the KP hierarchy:
*Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras*, by Kac, Raina, Rozhkovskaya.

- For a good introduction to the topic:
## Pierre Lairez,

*Why and how to calculate the period matrix of an algebraic variety*. [slides]*Abstract:*The period matrix of an algebraic variety is obtained by integrating differential forms over homology cycles. This matrix links the algebraic and topological aspects, which makes it possible to reveal some fine algebraic invariants, at least experimentally. The calculation of this matrix relies on a variety of classical symbolic and numerical tools: symbolic integration, numerical analytic continuation and path tracking. In this course, I will introduce these tools and provide an overview of recent and ongoing work related to period computations within the MATHEXP team.## Kilian Raschel,

*A functional equation approach to reflected Brownian motion*. [slides]*Abstract:*The last two decades have seen a flurry of activity around the combinatorial model of walks in the quarter plane. One reason for this is that this model not only occupies a central position in enumerative combinatorics (via bijections with other combinatorial models), but it has also motivated different communities (probability, complex analysis, functional equations, Galois theory, etc.) to work together to develop new techniques and derive new results. While it can be said that walks in the quarter plane are now fairly well understood, a natural question for a probabilist is to study the continuous analogue, namely Brownian motion in a quadrant. It turns out that this model was introduced forty years ago using intrinsically probabilistic methods. In this talk I will explain how delicate probabilistic problems associated with Brownian motion in cones can be solved using functional equations, in particular Galois results on the nature of the solutions to such equations. This talk is based on work with several co-authors, mainly arXiv:2101.01562 and arXiv:2401.10734. No prior knowledge of Brownian motion is required.

Talks (60 minutes):

## Marie Albenque,

*Sign clusters in random triangulations decorated by an Ising configuration*.*Abstract:*

In this talk, I will survey some recent results about Ising-decorated triangulations.

In this model, triangulations are sampled together with a spin configuration on their vertices (i.e. a 2-coloring of its vertices), with a probability biased by the number of monochromatic edges, via a parameter $\nu$. The fact that there exists a critical value for this model has been initially established in the physics literature by Kazakov and was rederived by combinatorial methods by Bousquet-Mélou and Schaeffer, and Bouttier, Di Francesco and Guitter. In this talk, I will give geometric evidence that this model undergoes a phase transition by studying its monochromatic clusters. In particular, we establish that, when $\nu$ is critical or subcritical, the cluster of the root is finite almost surely, and is infinite with positive probability for $\nu$ supercritical.

This is based on joint works with Laurent Ménard and with Laurent Ménard and Gilles Schaeffer.## Andrew Elvey Price,

*Refined enumeration of planar Eulerian orientations*. [slides]*Abstract:*I will discuss the enumeration of 4-regular planar maps decorated by an Eulerian orientation according to three natural parameters. Building on our previous work and work with Guttmann, we characterise the generating function counting these objects as the solution to a fairly simple but highly unusual system of functional equations. We solve this system in three two-parameter cases, one of which is the six vertex model on maps, which we previously solved in collaboration with Zinn-Justin, following a non-rigorous solution by Kostov. This is Joint work with Mireille Bousquet-Mélou.## Rafael Mohr,

*New algorithms for decomposing algebraic sets and polynomial ideals*. [slides]*Abstract:*

Algebraic sets, i.e., solution sets of polynomial systems of equations, model a wide variety of nonlinear problems, both in applied and pure mathematics. One of the most foundational results in algebraic geometry says that every algebraic set has a unique decomposition into*irreducible*algebraic sets and a frequent task is to decompose such an algebraic set into its irreducible components, or to produce some kind of coarser decomposition of the algebraic set in question. This task comes up for example in certain problems in robotics on the applied side and enumerative geometry on the pure side.

In this talk, we present a series of algorithms solving such decomposition problems for algebraic sets or, correspondingly, for polynomial ideals. All of these algorithms use Gröbner bases for polynomial ideals at their core.

Three of these algorithms produce so-called*equidimensional*decompositions of an algebraic set. They are designed to avoid potentially costly elimination operations and, partially, integrate*signature-based*Gröbner basis algorithms in a crucial way. Our software implementations of these algorithms will showcase their practical efficiency compared to state-of-the-art computer algebra systems. This is joint work with Christian Eder, Pierre Lairez and Mohab Safey El Din.

Another algorithm combines a Hensel lifting strategy with the*FGLM algorithm*to compute Gröbner bases for*generic fibers*of polynomial ideals. As such, it can be used for equidimensional or irreducible decomposition of an algebraic set or for primary decomposition of polynomial ideals, a task that comes up, for example, in a recent algorithm for*Whitney stratifications*by the speaker and Martin Helmer. This is joint work with Jérémy Berthomieu.## Dang-Khoa Nguyen,

*Height gaps for coefficients of D-finite power series and related results*. [slides]*Abstract:*

A power series $f(x_1,\ldots,x_m)\in \mathbb{C}[[x_1,\ldots,x_m]]$ is said to be D-finite if all the partial derivatives of $f$ span a finite dimensional vector space over the field $\mathbb{C}(x_1,\ldots,x_m)$. For the univariate series $f(x)=\sum a_nx^n$, this is equivalent to the condition that the sequence $(a_n)$ is P-recursive, meaning there exists a non-trivial linear recurrence relation of the form: $$P_d(n)a_{n+d}+\cdots+P_0(n)a_n=0$$ where the $P_i$'s are polynomials.

In this talk, we consider D-finite power series with algebraic coefficients and discuss the growth of the Weil height of these coefficients (this is joint work with Bell and Zannier). If time permits, we discuss how the method of our proof can be adapted to prove a new criterion for the so called Pólya-Carlson dichotomy and an application of this criterion to certain dynamical zeta functions.## Élie de Panafieu,

*Asymptotic expansion of regular graphs*. [slides]*Abstract:*A graph is regular if all its vertices have the same degree. Progress in the exact and asymptotic enumeration of regular graphs has involved a variety of techniques, including probabilistic methods, generating functions, and computer algebra. The generating function of $k$-regular graphs is known to be D-finite, and the associated differential equation has been computed for the first few values of $k$. We present a new exact formula enumerating graphs with their degrees constrained to belong to a given set. Our approach is based on generating function manipulations and extends to many graph variants, including bipartite graphs. Applying the Laplace method, we deduce the asymptotic expansion (unbounded number of error terms) of $k$-regular graphs, which was an open problem. Joint work with Emma Caizergues.## Marina Poulet,

*On the solutions of Mahler equations*. [slides]*Abstract:*Mahler equations are functional equations related to many areas such as automata theory. For example, the generating series of automatic sequences are solutions of such equations. In this talk, we will describe some properties of their solutions. We will first look at the form of the solutions when the Mahler equation is regular singular. In particular, we will explain how we can recognize this kind of Mahler equations, which is a joint work with Colin Faverjon. In a second part, we will be interested in the behavior of solutions which are meromorphic in the open unit disk, but not rational. More precisely, the unit circle is a natural boundary for these solutions and we will study the behavior of $f(z)$ when $z$ tends to $1^{-}$ radially, which is a joint work with Tanguy Rivoal.## Tanguy Rivoal,

*A Roth-type theorem for values of $E$-functions*. [slides]*Abstract:*I will report on the solution to a problem in the theory of Siegel’s $E$-functions initiated by Lang in the 60’s and considered in full generality by Chudnovsky in the 80’s: irrational values taken at rational points by $E$-functions with rational Taylor coefficients have irrationality exponent equal to 2. This result had been obtained before by Zudilin under stronger assumptions on algebraic independence of $E$-functions, satisfied by Bessel's function $J_0$ but not by all hypergeometric $E$-functions. This is a joint work with Stéphane Fischler (Université Paris-Saclay).## Sonia Rueda,

*Parametric factorization of algebro-geometric ODOs*. [slides]*Abstract:*

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). They allow the solvability of the spectral problem $Ly = λy$, for an algebraic parameter $λ$ and an algebro-geometric ODO $L$, whose centralizer is known to be the affine ring of a abstract spectral curve $Γ$. Their origin is linked to the Korteweg-de Vries hierarchy and to the seminal works on commuting ODOs by I. Schur and by Burchnall and Chaundy.

In this talk, we explore the computation of the defining ideal of a spectral curve. We will use differential resultants to effectively compute the defining ideal of the spectral curve, defined by the centralizer of a third order differential operator $L$, with coefficients in an arbitrary differential field of zero characteristic. Our results establish a new framework appropriate to develop a Picard-Vessiot theory for spectral problems. We use effective differential algebra to develop symbolic algorithms for the parametric factorization of algebro-geometric ODOs.

A parallel problem is the computation of new algebro-geometric ODOs, of order $≥ 3$. For this purpose we present new algorithms to compute bases of almost commuting operators and integrable hierarchies, based on the weighted differential monomialization introduced by G. Wilson in the eighties. I am presenting joint work with Maria-Ángeles Zurro, Antonio Jiménez Pastor, Rafael Hernández Heredero and Rafael Delgado as part of the project “Algorithmic Differential Algebra and Integrability” (ADAI), from the Spanish MICINN, PID2021-124473NB-I00.## Bruno Salvy,

*Positivity proofs for solutions of linear recurrences*. [slides + demo]*Abstract:*We show that for solutions of linear recurrences with polynomial coefficients of Poincaré type and with a unique simple dominant eigenvalue, positivity reduces to deciding the genericity of initial conditions in a precisely defined way. We give an algorithm that 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. This is joint work with Alaa Ibrahim.## Daniel Smertnig,

*Gaps in the growth of coefficients of Mahler functions*. [slides]*Abstract:*A formal power series in $z$ is a ($k$-)Mahler function if it satisfies a difference equation with respect to the endomorphism $z \mapsto z^k$. Such power series arise in transcendence theory (where they were introduced by Mahler) and as generating series of automatic, and more generally, regular sequences (in the sense of Allouche and Shallit). We show that Mahler functions with algebraic coefficients fall into five distinct classes, based on the asymptotic growth of their coefficients as measured by their logarithmic Weil height. One of these classes corresponds to automatic sequences, and two others to regular, but not automatic, sequences. (This is joint work with B. Adamczewski and J. Bell.)## Daniel Vargas Montoya,

*Congruences modulo $p$, algebraic independence and monodromy*. [slides]*Abstract:*Recently Adamczewski, Bell and Delaygue gave an algebraic independence criterion for power series with coefficients in $\mathbb Z$ that satisfy “$p$-Lucas congruences” for infinitely many prime numbers $p$. Most of the powers series satisfying this type of congruences are $G$-fonctions. In the first part of the talk, we are going to see how we can obtain this congruences when the power series is a solution of a differential operator. The main tools are, on the one hand, the $p$-adic study of the differential operator, strong Frobenius structure, and on the other hand, the classical notion of monodromy. In the second part, I introduce a new set of $G$-function, denoted MF, and show that the elements of MF also satisfy convenience congruences modulo $p$. Finally, we will see that in some cases these last congruences allow us to establish the algebraic independence of $G$-functions that are in MF.## Vitali Wachtel,

*Positive harmonic functions and Martin compactification*. [slides]*Abstract:*Positive harmonic functions play a rather important role in many problems in probability and in enumerative combinatorics. These functions can be seen as eigenfunctions for the corresponding transition kernel or as solutions to a fixed point equation. Accordingly, one has two typical mathematical issues: existence and uniqueness of the solution. To answer these questions one can a very nice and very powerful method: Martin compactification. In my talk I am going to explain some basic features of the Martin compactification and to give probabilistic and combinatorial examples, which illustrate the power of that method.## Harriet Walsh,

*Inhomogeneous random growth in half space and solutions of integrable equations*. [slides]*Abstract:*I will talk about models of two dimensional random growth (namely, polynuclear growth) which can be translated into probability laws on integer partitions by way of the RSK algorithm. As a consequence, we can study their asymptotic statistics with algebraic tools. I will focus on a model in half space with external sources driving growth at the edges, and present a new asymptotic distribution governing its interface fluctuations which interpolates between different universal Tracy-Widom distributions from random matrix theory, and encodes solutions of the Painlevé II integrable differential equation. Our approach uses connections between symmetric functions, matrix integrals, Hankel determinants, and a Riemann-Hilbert problem. Based on joint work with Mattia Cafasso, Alessandra Occelli and Daniel Ofner.

- Boris Adamczewski
- Jérémy Berthomieu
- Mireille Bousquet-Mélou
- Frédéric Chyzak
- Charlotte Hardouin
- Marc Mezzarobba
- Kilian Raschel
- Julien Roques

- The conference takes place at the Belambra club in Anglet.
- Meals included in the conference are from the dinner on Sunday 9 (19:00-20:30) to the lunch on Friday 14 (12:00-13:30).
- Arrivals are expected on Sunday 9, 17:00-20:00; the conference ends after the Friday lunch.
- Suggested transportation means:
- by train: stations in Biarritz or Bayonne are close, then take bus or taxi
- by plane: there is an airport in Biarritz, but international attendance may have to fly through Bordeaux or Paris (+ train)
- see also information by Belambra, in French

See the list of confirmed participants.