The parameter identifiability problem of a dynamical system is to determine whether the parameters of the system can be found from a given subset of variables of the system. Verifying whether the parameters are indentifiable is a necessary first step before a meaningful parameter estimation can take place. Reparametrization aims at improving identifiability properties of a dynamical system. We will discuss one of the approaches to do this, which is based on differential algebra and other symbolic computation methods. 

We consider the problem of symbolic integration int G(x,y(x)) dx where G is rational and y(x) is a non algebraic solution of a differential equation y'(x)=F(x,y(x)) with F rational. As y is transcendental, the Galois action generates a family of parametrized integrals I(x,h)=int G(x,y(x,h)) dx. We prove that I(x,h) is either differentially transcendental or up to parametrization change satisfies a linear differential equation in h with constant coefficients, called a telescoper. This notion generalizes elementary integration. We present an algorithm to compute such telescoper given a priori bound on their order and degree. For the specific foliation y=ln x, we present more complete analysis without need of a priori bound. As an application, we present an algorithm searching for Liouvillian solutions of a planar rational vector field.

We approach differential algebra and differential algebraic geometry from the point of view of topos theory and categorical logic, following the initial steps made by Keigher and Bunge in the early 80s. We consider the Keigher spectrum of a differential ring as the Hakim-Cole spectrum associated to the theory of differential local rings, which leads to the most explicit description of its structure sheaf as a sheaf on the Carrà-Ferro locale known thus far. 

In physics there often appear non-linear differential equations involving radical expressions. For such differential equations, most of the commonly used techniques fail. We present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives, or in the independent variables, into a system with polynomial relations among them by means of a rational change of variables. The solutions of the given equation and its transformation correspond one-to-one. For such algebraic differential equations more classical methods are available and often solutions can be derived. This is joint work with Rafael Sendra.

We consider chemical reaction networks (CRN) under mass-action kinetics. Those CRNs can be  modelled via ordinary differential equations. Conservation laws of CRNs are dependencies between the ODEs. Linear conservation laws are quite well-known and have been applied for model reduction, among others. However, polynomial conservation laws are not fully investigated. In this talk, we consider the problem of computing polynomial conservation laws under parameter change, that is, computing parametric polynomial conservation laws. We present an algorithm for computing parametric polynomial conservation laws of CRNs using comprehensive Gröbner systems and syzygies. We discuss benchmarking, experiments and computational issues of the algorithm.

This is a joint work with A. Desoeuvres, A. Iosif, C. Lüders, O. Radulescu, M. Seiß and T. Sturm.

A differential equation is strongly minimal if its solution set (taken in a universal differential field) is strongly minimal in the sense of model theory. In recent years, this model theoretic notion has been a powerful tool in the study of certain functional transcendence problems concerning the solutions of algebraic nonlinear differential equations. After describing this property and some of its consequences, I will explain a recent result which states that strong minimality is a “typical property’’ of an autonomous system of algebraic differential equations with high enough degree (or with sufficiently large poles).

During this talk, we will discuss the Schmidt–Kolchin conjecture on the vector space of differentially homogeneous polynomials in N+1 variables. Our interest in this question arouse from the (simple) observation that this vector space is isomorphic to the space of global sections of the direct sum of all twisted Green–Griffiths vector bundles on N-dimensional projective space. 

We will start by recalling the definition of differentially homogeneous polynomials, and state the conjecture. We will then briefly discuss the geometric correspondence described above. The remaining part of the talk will be devoted to the proof of the conjecture.

We will discuss a fundamental theorem for tropical differential equations analogue of the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al. and from Fink and Toghani, working only in the case of trivial valuation as introduced by Grigoriev, to differential equations with power series coefficients over any valued field. To do so, a crucial ingredient is the framework for tropical differential equations introduced by Giansiracusa and Mereta. As a corollary of the fundamental theorem, the radius of convergence of solutions of a differential equation over a nontrivially valued field can be computed tropically. This talk is based on results appearing in arXiv:2303.12124.

A differential alpha-resolvent of a monic univariate differential polynomial P is a way of "linearizing" P. A Powersum Formula (1997) computes these alpha-resolvents. When the coefficients of P are sufficiently differentially transcendental over constants, the alpha-resolvents have a well-defined homogeneous weight with respect to the coefficients of P. The Powersum Formula does not yield an alpha-resolvent of lowest weight. A method for factoring each term in the resolvent of 0-th and highest degree in alpha has been known since April 1999. In December 2022, a way of factoring the remaining terms had been discovered. We will demonstrate this factoring on a quadratic then a cubic.

The definition of resolvent can be generalized in different ways. The Powersum Formula also generalizes to these more general resolvents, yielding new interesting differential relations.

Consider the arc space of the fat point given by x^m = 0. This is defined by the quotient of k[x, x’, x’’, …] over the differential ideal generated by x^m . This algebra has a natural filtration by finite dimensional algebras corresponding to the truncation of arcs. We show that the dimensions of these algebras are given by the sequence m^{h+1}, where h is the order of truncation. Furthermore, we provide a full description of  the inverse system of the truncated ideals corresponding to those algebras in the case of double points. We link the elements of the inverse system to differentially homogeneous polynomials. 

This is a joint work with Gleb Pogudin.

 The foliated topology is a Grothendieck topology on the category of schemes endowed with a foliation. It is defined by analogy with the étale topology: roughly speaking, a foliated cover is a surjective local homeomorphism onto the leaves of the foliation. The goal of the talk is to explain how the foliated topology leads naturally to the notion of a foliated homotopy type of a differential field and to its higher differential absolute Galois groups. 

 In this talk, we will report on some recent results about linear Mahler equations. We will notably speak about automata, difference Galois theory, Hahn series. No prerequisite is required.