Galois groups for integrable and projectively integrable linear difference equations

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.