On the Definitions of Difference Galois Groups

TL;DR

This paper compares various definitions of difference Galois groups from algebra, analysis, and model theory, establishing their isomorphism over suitable fields and exploring properties of Picard-Vessiot extensions over fields with non-closed constant subfields.

Contribution

It demonstrates the equivalence of different Galois group definitions and investigates Picard-Vessiot extensions over more general fields.

Findings

Different definitions of difference Galois groups are isomorphic over suitable fields.

Properties of Picard-Vessiot extensions are studied over fields with non-closed constant subfields.

The paper bridges concepts across algebra, analysis, and model theory.

Abstract

We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of Picard-Vessiot extensions over fields with not necessarily algebraically closed subfields of constants.