Galois theories for $q$-difference equations: comparison theorems

TL;DR

This paper compares different parameterized Galois theories for q-difference equations, linking abstract differential properties of solutions to the behavior of meromorphic solutions, thereby completing prior work that lacked parameter considerations.

Contribution

It establishes comparison theorems connecting abstract parameterized Galois theories with meromorphic solutions for q-difference equations, extending previous results to include parameters.

Findings

Established link between abstract Galois theories and meromorphic solutions.

Completed the comparison of parameterized Galois theories for q-difference equations.

Proved that meromorphic solutions form a basis for linear q-difference equations with meromorphic coefficients.

Abstract

We establish some comparison results among the different parameterized Galois theories for $q$-difference equations, completing the work by CHatzidakis, Hardouin and Singer, that addresses the problem in the case without parameters. Our main result is the link between the abstract parameterized Galois theories, that give information on the differential properties of abstract solutions of $q$-difference equations, and the properties of meromorphic solutions of such equations. Notice that a linear $q$-difference equation with meromorphic coefficients always admits a basis of meromorphic solutions, as proven by Praagman.