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.

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.