# Differential Galois theory in Strasbourg

Europe/Paris
Salle séminaire (Strasbourg)

### Salle séminaire

#### Strasbourg

IRMA, université de Strasbourg
Description

Workshop on recent themes around differential Galois theory, computer algebra, and model theory.

List of speakers :

Guy Casale (Rennes)

Zoé Chatzidakis (ENS)

Frédéric Chyzak (INRIA Saclay)

Thierry Combot (Dijon)

Viktoria Heu (Strasbourg)

Loïc Teyssier (Strasbourg)

Changgui Zhang (Lille)

The conference is organized by Thomas Dreyfus (Strasbourg). If you wish to attend, please send him an e mail.

The workshop is founded by GDR EFI, CNRS ( PEPS JCJC funding) and IRMA.

• Wednesday, 4 September
• 14:00 15:00
Notions of difference closures of difference fields. 1h

It is well known that a differential field K of characteristic 0 is contained in a differential field which is differentially closed and has the property that it K-embeds in every differentially closed field containing K. Such a field is called a differential closure of K, and it is unique up to K-isomorphism. The difference closure is what model-theorists call a "prime model".

One can ask the same question about difference fields: do they have a difference closure, and is it unique? The immediate answer to both these questions is no, for trivial reasons: in most cases, there are continuum many ways of extending an automorphism of a field to its algebraic closure. Therefore a natural requirement is to impose that the field K be algebraically closed. Similarly, if the subfield of K fixed by the automorphism is not pseudo-finite, then there are continuum many ways of extending it to a pseudo-finite field, so one needs to add the hypothesis that the fixed subfield of K is pseudo-finite.

In this talk I will show by an example that even these two conditions do not suffice.

There are two (and more) natural strengthenings of the notion of difference closure, and we show that in characteristic 0, these notions do admit unique "closures" over any algebraically closed difference field K, provided the subfield of K fixed by the automorphism is large enough.

In characteristic p, no such result can hold.

All definitions will be introduced.

Speaker: Ms Zoé Chatzidakis
• 15:00 15:30
Break 30m
• 15:30 16:30
Integration on Darbouxian foliations 1h

Consider a Darbouxian function $f=F_0+\sum \lambda_i \ln F_i$ with $F_i$ rational functions in two variables, and the foliation of curves $\mathcal{C}_h=\{f(x,y)=h\}$. We consider the problem of symbolic integration of a rational function $G$ along $\mathcal{C}_h$. If the monodromy of the integral satisfies a differential equation in $h$, then it is linear with constant coefficients, and the integral can be expressed in terms of Liouvillian functions restricted to $\mathcal{C}_h$. Such situation is exceptional, but is however more general than elementary integration. We present an algorithm to test the existence of such differential equation and return the Liouvillian expression of the integral if it exists.

Speaker: Mr Thierry Combot
• Thursday, 5 September
• 09:30 10:30
Quotients and equations 1h

Quotients are ubiquitous in Mathematics, and a general question is whether a certain category of sets allows quotients. For the category of definable sets in a given structure, the model theoretic approach is called elimination of imaginaries. For algebraically closed fields, Chevalley’s theorem and the existence of a field of definition of a variety imply that a quotient of a Zariski constructible set by a Zariski constructible equivalence relation is again constructible. Similar results hold for other classes of fields, such as differentially closed fields.
In this talk, we will focus on separably closed fields and differentially closed fields of positive characteristic. In joint work with Martin Ziegler, we will provide a natural expansion of the language to achieve elimination of imaginaries, by showing that these theories are equational. Equationality, introduced by Srour, and later considered by Srour and Pillay, is a generalisation of local noetherianity. We will present the main ideas of the proof, without assuming a deep knowledge of model theory.

Speaker: Martin Martin-Pizarro
• 10:30 11:00
Break 30m
• 11:00 12:00
On Miyake's algorithm for formal reduction of linear differential systems 1h
• 14:00 15:00
The Riemann-Hilbert mapping in genus two 1h
Speaker: Viktoria Heu
• 15:00 16:00
Becker’s conjecture on Mahler functions 1h

In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z) = 1$. In this work, we prove Becker’s conjecture in the best possible form; we show that the rational function $R(z)$ can be taken to be a polynomial $z^γ Q(z)$ for some explicit non-negative integer $γ$ and such that $1/Q(z)$ is $k$-regular. (This is joint work with Jason P. Bell, Michael Coons, and Philippe Dumas.)

Speaker: Frédéric Chyzak
• 16:00 16:30
Break 30m
• 16:30 17:30
Inverse problem for germs of parabolic diffeomorphisms of the complex line 1h
Speaker: Loïc Teyssier
• Friday, 6 September
• 09:30 10:30
Ax-Lindemann-Weierstrass theorem for the genus 0 Fuchsian groups 1h
Speaker: Mr Guy Casale
• 10:30 11:00
Break 30m
• 11:00 12:00
The asymptotic proprieties of solutions of a q-difference equation as q tends to a root of unity 1h
Speaker: Changgui Zhang
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