On the computation of the difference Galois groups of order three equations

TL;DR

This paper develops methods to compute difference Galois groups for third-order equations across various difference operators, providing criteria for irreducibility, imprimitivity, and conditions for differential transcendence of solutions.

Contribution

It introduces a unified approach to compute difference Galois groups for order three equations in multiple difference operator cases, including new criteria and an explicit example.

Findings

Criteria for irreducible Galois groups

Criteria for imprimitive Galois groups

Conditions for differential transcendence of solutions

Abstract

In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the $q$-difference operator (Case Q), the Mahler operator (Case M) and the elliptic case (Case E). We show that the general problem can be reduced to several ancillary problems. We prove criteria to detect the irreducible and imprimitive Galois groups. Finally, we give a sufficient condition of differential transcendence of solutions of order three difference equations. We also compute the difference Galois group of an equation suggested by Wadim Zudilin.