Galois groups for integrable and projectively integrable linear difference equations

TL;DR

This paper characterizes the Galois groups of integrable and projectively integrable linear difference equations over complex rational functions, showing they are solvable and providing criteria for hypertranscendence.

Contribution

It applies recent theoretical results to classify Galois groups of such systems and establishes their solvability, along with hypertranscendence conditions.

Findings

Galois groups are solvable for integrable systems

Characterization of Galois groups for different difference operators

Hypertranscendence criteria for solutions

Abstract

We consider first-order linear difference systems over $\mathbb{C}(x)$, with respect to a difference operator $\sigma$ that is either a shift $\sigma:x\mapsto x+1$, $q$-dilation $\sigma:x\mapsto qx$ with $q\in{\mathbb{C}^\times}$ not a root of unity, or Mahler operator $\sigma:x\mapsto x^q$ with $q\in\mathbb{Z}_{\geq 2}$. Such a system is integrable if its solutions also satisfy a linear differential system; it is projectively integrable if it becomes integrable "after moding out by scalars." We apply recent results of Sch\"{a}fke and Singer to characterize which groups can occur as Galois groups of integrable or projectively integrable linear difference systems. In particular, such groups must be solvable. Finally, we give hypertranscendence criteria.