The study of generating series in enumerative combinatorics or of stationary distribution and their Laplace transforms in probabilities yield to strange functional equations, derived from a step by step analysis in Combinatorics or directly from Ito's formula in Probability. Associated to these functional equations, one finds an algebraic curve or a surface defined over a function field. This geometric object is called the kernel of the functional equation. Evaluating the functional equation on the kernel allows to deduce a linear dynamical equation of order one defined over the kernel. Differential Galois theory allows then to build a dictionary between some dynamical properties of points on the curve and the differential algebraic behavior of the initial generating series and probabilistic objects. In this very informal talk, I will try to present a brief history of these subjects and the different domains of mathematics which interplay in these studies. This talk is connected to collaborations with Thomas Dreyfus, Julien Roques and Michael Singer as well as some recent works with Mireille Bousquet-Mélou, Andrew Elvey Price, Sandro Franceschi and Kilian Raschel.

Theories of existentially closed fields with operators play an important role in the algebraic model theory and in the development of the general model theory. Stable fields with operators have already proved their importance outside of model theory, there exists a well-rounded model theory of simple fields with operators, and now we are approaching a wilder forest of non-simple fields with operators, which are still tame in some sense.To produce new examples of such fields, i.e. non-stable in a controlled manner, we involve a group action by automorphisms (here - a group action by differential automorphisms). My plan for this talk is to outline how this general procedure works and to present one new example of a theory of existentially closed fields with operators, more precisely - fields with finitely many commuting derivations and finitely many differential automorphisms which correspond to a group. The example comes from a joint work with Omar León Sánchez, which you may find here: arxiv.org/abs/2012.14376

Many stream or block ciphers of application interest such as Trivium, Keeloq, LFSR systems with combiner (E0 of Bluetooth), etc can be modeled as systems of explicit ordinary difference equations over finite fields. Such systems indeed determine the evolution over time of the internal registers of these "difference ciphers". The use of the formal theory of difference equations allows the study of some fundamental properties of these ciphers, such as their invertibility, and the correct definition of algebraic attacks for the purpose of assessing their security. Such modeling and the corresponding cryptanalysis allows hence the development of new ciphers.

Many reconstruction algorithms from moments of algebraic data were developed in optimization, analysis or statistics. Lasserre and Putinar proposed an exact reconstruction algorithm for the algebraic support of the Lebesgue measure, or of measures with density equal to the exponential of a known polynomial. Their approach relies on linear recurrences for the moments, obtained using Stokes theorem. In this talk, we discuss an extension of this study to measures with holonomic densities (i.e. they satisfy a holonomic system of linear partial or ordinary differential equations with polynomial coefficients) and support with real algebraic boundary. Using Stokes theorem, our approach computes recurrences for the moments which stay linear w.r.t the coefficients of the polynomial vanishing over the support boundary. This property allows for an efficient reconstruction method (from sufficiently many moments) for both these coefficients and those of the polynomials involved in the holonomic operators which annihilate the density, by solving linear systems only. This is a joint work with Florent Bréhard and Jean-Bernard Lasserre.

Abstract in pdf

I'll discuss some joint work, still in progress, with Gal Binyamini, Harry Schmidt, and Margaret Thomas, in which we prove effective forms of the Pila-Wilkie Theorem for various structures involving pfaffian functions. I'll also discuss applications to effective results on some unlikely intersection problems.

A first-order algebraic difference equation has the form F(x,y,σ(y))=0 for some polynomial F and difference operator σ. We propose an algebro-geometric approach for studying rational solutions of the given difference equation. In case F is independent of x and σ is the shift operator, we give an upper bound for the degrees of the rational solutions, and thus derive a complete algorithm for computing corresponding rational solutions.

Let R(x) in C(x) be a rational function over the complex numbers and S(y) = (y''/y')' -1/2(y''/y')² be the Schwarzian derivative of y with respect to the variable x. Kummer equation is S(y) +R(y) (y')² = R(x). This equation appears in different works, from original particular case of Kummer, to Lie and Cartan theory of Lie pseudogroups, ODE approach to uniformisation, Ritt hypertranscendence theorem, Ecalle's synthesis of binary diffeomorphisms ...

In this talk I will present some properties of this equation from algebraic pseudogroups point of view. Examples with finitely many algebraic solutions and examples with a "Kolchin dense" set of algebraic solutions will be given. A complete description of equations with Kolchin dense set of algebraic solution would give a nice generalisation of Ritt hypertranscence theorem, describing the solutions of algebraic difference equation P(f(qz),f(z)) = 0 satisfying a differential equation (here P in C[X,Y], |q| > 1).

Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case there are periodic tropical recurrent sequences which are similar to classical linear recurrent sequences. A question is studied when there exists a non-periodic tropical recurrent sequence satisfying a given vector, and partial answers are provided to this question. Also an algorithm is designed which tests existence of non-periodic tropical recurrent sequences satisfying a given vector with integer coordinates. Finally, we introduce a tropical entropy of a vector, provide some bounds on it and extend this concept to tropical multivariable recurrent sequences.

This talk is devoted to a parametric study of Liouvillian solutions of the general trace-free second order linear differential equation with a Laurent polynomial coefficient of arbitrary orders at 0 and infinity. It is known that, for an algebraic family of linear differential equations with rational coefficients, the subfamily of differential equations whose solution space is Gal-isomorphic to a given representation of some fixed subgroup containing its Galois group and fixed exponents at the singularities, is a constructible subfamily. In this case (with arbitrary exponents), by means of a combination of Kovacic's algorithm and asymptotic iterations, we will see that the subfamily of integrable cases is a singular analytic submanifold, and an enumerable union of algebraic varieties. We will compute explicitly the equations of its irreducible components by an iterative method. This talk is based on a joint work with P. B. Acosta Humanez and H. Venegas Gómez.

Abstract in pdf

The model theory of H-fields, introduced by van den Dries and Aschenbrenner, provides a general framework to study differential equations in ordered fields. This theory admits in particular geometric, formal, and number theoretic models: Hardy fields, which are fields of differentiable real-valued germs at infinity, transseries, which are formal series involving exponentials and logarithms, and surreal numbers, which are abstract ordered quantities that can mimic both germs and transseries. I will give an overview of the theory, these different types of models and their connections.

A conjecture of Zilber, motivated by the model theory of complex exponentiation, predicts the existence of many intersections between the graph of the exponential function (extended pointwise to several variables) and algebraic varieties, as long as the varieties satisfy some geometric conditions related to Schanuel's conjecture. Some instances are known to be true, most notably when the projections of the varieties onto the domain side of the graph have maximal dimension, i.e. equal to the number of variables (by work of Brownawell-Masser and D'Aquino-Fornasiero-Terzo).

I will discuss the case of varieties whose projection on the domain has dimension one, that is, it is a curve. Then such intersections always exist if the curve is not contained in a translate of a rational hyperplane (and if it is, a trivial geometric condition determines when there are no intersections). We can prove this by appealing to the classical theory of differentials on compact Riemann surfaces and a suitable instance of the Ax-Lindemann-Weierstrass theorem. This is joint work with David Masser.

Abstract: Set positive invariance is an important concept in the theory of dynamical systems and one which also has practical applications in areas of computer science, such as formal verification, as well as in control theory. Great progress has been made in understanding positively invariant sets in continuous dynamical systems and powerful computational tools have been developed for reasoning about them; however, many of the insights from recent developments in this area have largely remained folklore and are not elaborated in existing literature. This presentation contributes an explicit development of modern methods for checking positively invariant sets of ordinary differential equations and describes two possible characterizations of positive invariants: one based on the real induction principle, and a novel alternative based on topological notions. The two characterizations, while in a certain sense equivalent, lead to two different decision procedures for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations.

Abstract: Roughly speaking, the essential dimension of an algebraic object counts the number of parameters needed to describe the object. In this talk, we define an analogue of essential dimension in differential Galois theory. As application, we show that the number of coefficients in a general homogeneous linear differential equation over a field cannot be reduced via gauge transformations over the given field. We also give lower bounds on the number of parameters needed to write down certain generic Picard-Vessiot extensions.