Iterative q difference Galois Theory

TL;DR

This paper develops a Galois theory for q-difference equations when q is a root of unity, extending and generalizing existing theories to include iterative differential equations over fields of positive characteristic.

Contribution

It introduces a novel Galois theory for q-difference equations at roots of unity, unifying and broadening previous frameworks.

Findings

Generalizes Galois theory of q-difference equations

Connects q-difference and iterative differential equations

Provides a new algebraic framework for positive characteristic fields

Abstract

We propose in this paper a Galois theory of $q$-difference equations where q is a root of unity. This theory is the q difference analogue of the Galois theory of iterative differential equations, that is differential equations over fields of positive characteristic. This theory contains and generalizes the Galois theory of q difference equations developed by Singer and van der Put.