Summability has been a central object of study in difference algebra over the past half-century. It serves as a cornerstone of algebraic methods to study linear recurrences over various fields of coefficients and with respect to various kinds of difference operators. Recently, Dreyfus, Hardouin, Roques, and Singer introduced a notion of elliptic orbital residues, which altogether serve as a partial obstruction to summability for elliptic functions with respect to the shift by a non-torsion point over an elliptic curve. We explain how to refine this into a complete obstruction, which promises to be useful in applications of difference equations over elliptic curves, such as elliptic hypergeometric functions and the combinatorics of walks in the quarter plane. This is joint work with Carlos Arreche.

Sometimes, solutions to a system of differential equations can be considered as a subset of the constants of some differentially closed field of characteristic zero together with some fixed solutions. When this happens, we say the set of generic solutions is internal to the constants of this field. This talk describes progress, from joint work with Léo Jimenez, on developing an algebraic criterion for when solutions sets are almost internal to the constants.

Taking the reduction of a D-finite power series modulo a prime number, one often obtains an algebraic power series. For example, for diagonals of multivariate rational functions this was observed by Furstenberg in 1967, and for hypergeometric functions this is the content of recent results by D. Vargas-Montoya. In these cases, one can consider the (usual, algebraic) Galois group of the reduction over the field of rational functions.

In this talk, I will showcase many examples of D-finite series of different nature for which we are (almost) able to compute said Galois groups for all prime numbers for which they are defined. I will then collect these observations to raise questions on the general behavior of these groups: Is there some uniformity of the Galois groups of the reductions of a given D-finite series with respect to the prime number, and does there exist an object in characteristic zero governing their behavior, like the differential Galois group of the minimal equation for the series?

This talk is based on ongoing joint work with X. Caruso and D. Vargas-Montoya.

In 1852, Eisenstein proved that Taylor expansions of algebraic functions have strong integrality properties. I'll explain a strengthening of the Grothendieck-Katz p-curvature conjecture that proposes that these properties characterize algebraic functions among local solutions to (possibly non-linear!) algebraic differential equations at non-singular points. We can verify this conjecture in a number of cases of algebro-geometric interest, in both linear and non-linear settings. This is joint work with Josh Lam.

During the period between 1833 and 1841, Liouville introduced the class of elementary functions to obtain analogues of the notion of resolubility by radicals for transcendental and differential equations. Can the primitive of an algebraic function be expressed as an elementary function? Does a planar vector field admit an elementary first integral?  Is the restricted (real-analytic) cosine function definable in the structure (R,+,x, exp)?

In my talk, I will describe how the (omega-stable) theory of blurred exponential fields axiomatized by Kirby around 2007 provides a new framework for the development of model-theoretic techniques to unify and study the various notions of integrability coming from Liouville's class of elementary functions.

This is joint work with Jonathan Kirby.

For a polynomial dynamical system, we study the problem of computing the minimal differential equation satisfied by a chosen coordinate. We give a bound for the Newton polytope of the support of such an equation and show that our bound is sharp in ``more than half of the cases’'. We show the algorithm based on this bound and demonstrate that our implementation of the algorithm can tackle problems which are out of reach for the state-of-the-art software for differential elimination.

A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equation is called an algebraic difference equation (ADE). 

This talk is related to recent investigations by Ovchinnikov, Pogudin, Scanlon, Thieu Vo, and Wibmer for solving systems of difference equations in sequences. Building upon R. Cohn's results (1948) on solutions of ordinary ADEs, we explore D-algebraic sequences and discuss some of their closure properties.

A power series is said to be D-finite ("differentially finite”) if it satisfies a linear differential equation with polynomial coefficients. D-finite power series are ubiquitous in combinatorics, number theory and mathematical physics. In his seminal article on D-finite functions [S1], Richard P. Stanley asked for "an algorithm suitable for computer implementation" to decide whether a given D-finite power series is algebraic or transcendental.  Although Stanley insisted on the practical aspect of the targeted algorithm, at the time he formulated the problem it was unknown whether the task of recognizing algebraicity of D-finite functions is decidable even in theory. We first prove such a decidability result. The corresponding algorithm has too high a complexity to be useful in practice. This is because it relies on the costly algorithm from [S2], which involves, among other things, factoring linear differential operators, solving huge polynomial systems and solving Abel’s problem. I will then present an answer to Stanley’s question based on "minimization" of linear differential equations, and illustrate it through examples coming from combinatorics and number theory. (Work in collaboration with Bruno Salvy and Michael F. Singer.)

[S1] R. P. Stanley, Differentiably finite power series. European J. Combin. 1 (1980), no. 2, 175–188.

[S2] M. F. Singer, Algebraic solutions of nth order linear differential equations. Proc. Queen’s Number Theory Conf. 1979, Queen's Papers in Pure and Appl. Math., 54 (1980), 379–420.

The Jacobi Bound Conjecture gives an upper bound on the number of constants of integration needed to describe a general nonlinear system of ordinary differential equations. It states that if u1,...,un are algebraic ODEs in dependent variables x1,x2,..,xn then the absolute dimension (Krull dimension) of the differential variety of a component of finite absolute dimension is bounded by the tropical determinant (max,+) of the order matrix of the equations. We will give an introduction to this problem (including the history and some proofs) and will discuss some joint work in progress with David Zureick-Brown related to this problem.

In this talk, we considered the problem of estimating parameters based on Physics-Informed Neural Networks (PINNs), which are types of deep neural networks integrating governing equations behind the data. In general, it is known that learning PINNs to estimate unknown parameters given partial observations is challenging. Based on this, we propose to introduce differential elimination to the inverse problem of PINNs. The validity of the proposed method is demonstrated through numerical experiments using specific examples of epidemiological models.

In this talk we consider the problem of automatically proving inequalities involving sequences that are only given in terms of their defining recurrence relations. We will consider sequences satisfying linear recurrences with constant coefficients (C-recursive), linear recurrences with polynomial coefficients (P-recursive or holonomic), or certain systems of polynomial non-linear recurrences (admissible). Even when restricting to the simplest class, C-recursive, and the positivity problem, decidability is only known for orders up to five. And yet, there are computer algebra methods that try to tackle this problem. Obviously, they do not succeed on all types of input and so even though correctness can be proven, termination is typically an issue. In this talk, we will give an overview on this topic, share some available methods and showcases where algorithmic proofs actually succeeded.