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DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

Part of: Differential and difference algebra Differential equations in the complex domain

Published online by Cambridge University Press:  17 April 2015

Lucia Di Vizio ,
Charlotte Hardouin  and
Michael Wibmer
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])
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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.

Keywords

difference algebraGalois theoryalgebraic differential equationsdiscrete isomonodromy

MSC classification

Primary: 12H10: Difference algebra 12H20: Abstract differential equations 34M15: Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) 33C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 1 , February 2017 , pp. 59 - 119
Copyright
© Cambridge University Press 2015 

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