Mahler discrete residues and summability for rational functions

TL;DR

This paper introduces Mahler discrete residues for rational functions, providing a complete criterion for Mahler summability and extending concepts from shift and q-difference cases to Mahler difference equations.

Contribution

It develops Mahler discrete residues as a new tool to determine Mahler summability of rational functions, extending previous residue theories to the Mahler setting.

Findings

Mahler residues form a complete obstruction to Mahler summability.

Extension of discrete and q-discrete residues to Mahler case.

Framework for addressing telescoping and Galois theory problems in Mahler difference fields.

Abstract

We construct Mahler discrete residues for rational functions and show that they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p > 1$. This extends to the Mahler case the analogous notions, properties, and applications of discrete residues (in the shift case) and $q$-discrete residues (in the $q$-difference case) developed by Chen and Singer. Along the way we define several additional notions that promise to be useful for addressing related questions involving Mahler difference fields of rational functions, including in particular telescoping problems and problems in the (differential) Galois theory of Mahler difference equations.