On σδ-Picard–Vessiot Extensions

Abstract

We study the differential Galois theory of difference equations under weaker hypothesis on the field of σ-constants. This framework yields a new approach to results by C. Hardouin and M. Singer, which answers positively 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.

Key Words:

• Differential and difference algebra
• Differential Galois theory
• Model theory

2000 Mathematics Subject Classification:

• Primary 12H05, 12H10, (12H99)
• Secondary 12L12

ACKNOWLEDGMENT

I wish to thank my advisor, Zoé Chatzidakis, for invaluable discussions and support, both personal and academic, during all this time.

Notes

¹Here Σ and Π are the sets of automorphisms and linear derivations, respectively, appearing in the equations under study, and Π is a set of arbitrary derivations, the requirements on which are just commutativity with Δ ∪ Σ.

Communicated by D. Macpherson.