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

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

2011-03-25

arxiv.org/abs/1103.5067v2

Mathematics

Quantum Algebra

17 pages. To appear in Pacific Journal of Mathematics

Scientific paper

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.

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