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DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

Published online by Cambridge University Press:  17 April 2015

Lucia Di Vizio
Affiliation:
Laboratoire de Mathématiques UMR8100, UVSQ, 45 avenue des États-Unis, 78035 Versailles cedex, France ([email protected])
Charlotte Hardouin
Affiliation:
Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex 9, France ([email protected])
Michael Wibmer
Affiliation:
Lehrstuhl für Mathematik (Algebra), RWTH Aachen, 52056 Aachen, Germany ([email protected])

Abstract

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

Type
Research Article
Copyright
© Cambridge University Press 2015 

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