On sigma-delta-Picard-Vessiot extensions

TL;DR

This paper advances the differential Galois theory for difference equations by weakening the constant field assumptions, confirming existing results under classical conditions, and explicitly determining the Galois group's structure.

Contribution

It introduces a new approach to differential Galois theory of difference equations under weaker constant field hypotheses, extending and confirming previous results.

Findings

Galois group is isomorphic to classical results over a suitable field

Explicit calculation of the Galois group's connected components

Validation of known results under relaxed hypotheses

Abstract

We study the differential Galois theory of difference equations under weaker hypothesis on the field of constants of the automorphism. This framework yields a new approach to results by C.Hardouin and M.Singer, which answers possitively a question by M.Singer: under the classical hypothesis, the known results are still valid. In particular, our Galois group is isomorphic to theirs over a suitable field. We also explicitly calculate the number of connected components of the Galois group.