Descent for differential Galois theory of difference equations.   Confluence and q-dependency
  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. 
Lucia DI Vizio and Charlotte Hardouin
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