Descent for differential Galois theory of difference equations: confluence andq-dependence
Lucia Di Vizio, Charlotte Hardouin · Pacific Journal of Mathematics · 2012
The present paper contains two results that generalize and improve constructions of Hardouin and Singer.In the case of a derivation, we prove that the parametrized Galois theory for difference equations constructed by Hardouin and Singer 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 dependence with respect to x d d x and q d dq .We show that the parametrized 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.