Descent for differential Galois theory of difference equations. Confluence and q-dependency

Abstract

The present paper essentially contains two results that generalize and improve some of the constructions of [arXiv:0801.1493]. First of all, in the case of one derivation, we prove that the parameterized Galois theory for difference equations constructed in [arXiv:0801.1493] can be descended from a differentially closed to an algebraically closed field. In the second part of the paper, we show that the theory can be applied to deformations of q-series, to study the differential dependency with respect to x\frac{d}{dx} and q\frac{d}{dq}. We show that the parameterized difference Galois group (with respect to a convenient derivation defined in the text) of the Jacobi Theta function can be considered as the Galoisian counterpart of the heat equation.

17 pages. To appear in Pacific Journal of Mathematics

References in corpus (4)

• Parameterized Picard-Vessiot extensions and Atiyah extensions
• Existence of $\partial$-parameterized Picard-Vessiot extensions over fields with algebraically closed constants
• A Chevalley theorem for difference equations
• Intrinsic approach to Galois theory of q-difference equations, with the preface to Part 4 "The Galois D-groupoid of a q-difference system'' by Anne Granier

Cited by in corpus (2)

• Existence of $\partial$-parameterized Picard-Vessiot extensions over fields with algebraically closed constants
• Residues and Telescopers for Rational Functions