Given a polynomial derivation D, we study the existence of (algebraic) particular integrals, known as Darboux polynomials or eigenpolynomials (said differently, the principal ideal generated by a Darboux polynomial is differential for D). A straightforward and effective method is ansatz-based: one fixes a bound on the total degree of an ansatz p, and solves for all the relations that the undetermined coefficients of p have to satisfy for the normal form of D(p) with respect to p vanishes. If successful such computation either provides a candidate Darboux polynomial or proves the non-existence of such object up to the fixed degree bound. We build upon this simple idea with an important twist: we drop the bound on the degree or, more precisely, we manipulate objects where both the coefficients and the multi-degree itself are undetermined. This framework not only permits to reproduce already known non-existence proofs but allowed us to provide (constructive) proofs for open conjectures about the existence of Darboux polynomials for some dynamical systems like the Shimizu-Morioka model for the convection of turbulent flows. In this talk we will present the most recent theoretical and practical advances achieved in this direction.

We prove an Ax-Lindemann-Weierstrass differential transcendence result for Euler's gamma function, namely that the functions Γ(ν-ζ1(ν)),…, Γ(ν-ζn(ν)) are differentially independent over the field of rational functions in the variable ν, with coefficients in the field k of 1-periodic meromorphic functions over the complex numbers, as soon as ζ1,…, ζn determine a set of algebraic functions over k, stable by conjugation and pairwise distinct modulo the integers. 

To prove this result we use both the Galois theory of difference equations and the theory of a characteristic zero analog of the Carlitz module introduced by the second author in 2013. As an intermediate result we give an explicit description of the Picard-Vessiot rings and of the Galois groups associated to the operators in the image of the Carlitz module, using techniques inspired by the Carlitz-Hayes theory. This is a joint work with Federico Pellarin. 

Consider a subfield of the field of rational functions. It is finitely generated, and a natural question is to find a simple generating set. In this talk, we will present an algorithm to do this. We will also discuss one practical application: the problem of structural parameter identifiability of ODEs. Joint work with Gleb Pogudin.

Difference algebraic groups occur naturally as Galois groups in certain Galois theories, and they can also be used as a tool to study difference Hopf structures. In this talk I will discuss some finiteness properties of partial difference algebraic groups, in particular the existence of a dimension polynomial for any partial difference algebraic group. This allows us to define some invariants of a difference algebraic group such as its difference type and difference dimension.

A rational dynamical system consists of a variety X and a rational (algebraic) map from X to itself. We study a special class of such dynamical systems, the isotrivial ones, showing for example that if the system has no "first integrals", then it has only finitely many maximal proper invariant subvarieties (the Dixmier-Moeglin problem). This result, as well as others, is a corollary of the main theorem, which states that the birational automorphism group of such a system is an algebraic group. This, in turn, is a special case of the "binding group" theorem in model theory. I will explain the definitions and statements involved, as well as the model theoretic point of view on the situation. This is joint work with Rahim Moosa from the University of Waterloo.

Classically, differential Galois theories, including Picard-Vessiot theory for linear differential equations and Kolchin's strongly normal theory, associate an algebraic group to a differential field extension. By using categorical Galois theory and novel techniques of descent, we develop a differential Galois theory that applies to morphisms of differential schemes. We find effective criteria for universality of geometric quotients and thus pave the way to numerous applications of the theory. In particular, we show how it can be used to treat differential equations parametrised over a scheme (including problems of commuting differential operators seen as parametrised over associated spectral curves), and to study Galois groupoids associated to linear differential equations.

Linear differential equations with polynomial coefficients in a complex variable appear naturally in fields such as combinatorics, algebraic geometry, and mathematical physics. Their numerical solution, and in particular the numerical solution of connection problems between regular singular points, finds applications that include the high-precision computation of values of Feynmann integrals, volumes of semi-algebraic sets, or more generally periods of algebraic varieties, the asymptotic expansion of linearly recurrent sequences, and the factorization of linear differential operators. In this talk, I will give a demonstration of a numerical ODE solver that focuses on this very special class of equations, supports arbitrary-precision computations, and provides rigorous error bounds along with the numerical results. I will also demonstrate software based on this solver for some of the applications listed above.

This talk presents recent results on the reconstructibility and identifiability of nonlinear dynamical networks. These results rely on the identification of specific variables, called variables of interest, together with local differential relations obtained through elimination procedures applied to the equations governing each node. This framework makes it possible to propagate reconstructibility and identifiability properties across the network structure. As an application, I illustrate the approach on the neuronal network of C. elegans involved in chemotaxis.

In combinatorics, and more specifically in the study of lattice walks confined to cones such as the quarter plane, it has become standard to investigate the differential nature of the associated generating functions. When these functions are differentially finite or differentially algebraic, one can derive concrete information about the model, including asymptotics, recurrences, and, in some cases, closed-form expressions. By contrast, the combinatorial significance of differential transcendence remains much less understood.

In this talk, I will present a family of examples for which we provide a combinatorial and probabilistic interpretation of differential transcendence. Focusing on singular walks in the quarter plane, we show that, while the generating function is generically differentially transcendental, it exhibits a stronger form of differential transcendence at the spectral radius. This phenomenon is explained by an underlying probabilistic critical behavior.

This is joint work with Alin Bostan and Lucia Di Vizio (arXiv:2504.13542).

In classical Galois theory a finite Galois extension can be equivalently defined as either a finite separable normal extension or as the splitting field of a separable polynomial. In the setting of Galois theory of differential fields, Kolchin’s notion of a strongly normal extension plays the role of a separable normal extension.  The concluding theorem of Kolchin’s 1973 book Differential Algebra and Algebraic Groups characterizes strongly normal extensions as exactly those differential extensions generated by solutions to logarithmic differential equations on torsors for algebraic groups defined over the constant field of the base which introduce no new constants. We describe recent joint work with Omar León Sánchez giving a complete generalization of the result in the parameterized multiple derivations setting, namely, characterizing generalized strongly normal differential Galois extensions as being no-new-constants extensions generated by solutions to logarithmic differential equations on parameterized D-torsors for the differential Galois group. This builds on a long process of partial generalizations. We give a cohomological characterization of when the logarithmic differential equation can be found on the Galois group instead of on a nontrivial torsor. Along the way, we give a generalization to the parameterized setting of Kolchin’s result that for an algebraic group defined over a differential field there is a canonical bijection between (the isomorphism classes of) algebraic and differential algebraic torsors. We also give a model theoretic generalization of the main theorem and describe examples both extant and missing from the literature.