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 $q\in {\mathbb{Z}}_{\ge 2}$. 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 Schäfke 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.

References in corpus (4)

• Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
• Isomonodromic differential equations and differential categories
• Tannakian approach to linear differential algebraic groups
• Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

Cited by in corpus (8)

• Hypertranscendence and linear difference equations
• Computing differential Galois groups of second-order linear $q$-difference equations
• Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
• Algebraic independence and linear difference equations
• Differential transcendence of solutions of difference Riccati equations and Tietze's treatment
• Twisted Mahler discrete residues
• First-order factors of linear Mahler operators
• Hypertranscendence and linear difference equations, the exponential case