Vol. 256, No. 1, 2012

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Descent for differential Galois theory of difference equations: confluence and q-dependence

Lucia Di Vizio and Charlotte Hardouin

Vol. 256 (2012), No. 1, 79–104

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 xd-
dx 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.

linear difference equations, Galois theory, hypertranscendence
Mathematical Subject Classification 2010
Primary: 12H99, 39A13, 39A99
Received: 25 March 2011
Revised: 25 November 2011
Accepted: 29 November 2011
Published: 6 May 2012
Lucia Di Vizio
Laboratoire de Mathématiques, bâtiment Fermat
Université de Versailles-St Quentin
45 avenue des États-Unis
78035 Versailles Cedex
Charlotte Hardouin
Institut de Mathématiques de Toulouse
Université Paul Sabatier
118 route de Narbonne
31062 Toulouse Cedex 9