Deciding summability via residues in theory and in practice
Carlos E. Arreche
TL;DR
This paper surveys the summability problem in difference fields through the lens of discrete residues, outlining how these obstructions govern solvability for various difference operators. It surveys four concrete settings—shift, q-dilation, Mahler, and elliptic—providing precise residue definitions, summability criteria (vanishing residues), and current algorithmic prospects. In the shift case, practical representations (e.g., B_k, D_k) enable efficient residue computation without full partial fraction decompositions, while q-dilation and Mahler settings present deeper structural challenges and partial results, including twisted Mahler variants. In the elliptic setting, orbital residues underpin partial obstruction theory, with panororbital residues and ongoing work toward comprehensive algorithms; collectively, these obstructions underpin practical tools for creative telescoping and difference-Galois theory computations.
Abstract
In difference algebra, summability arises as a basic problem upon which rests the effective solution of other more elaborate problems, such as creative telescoping problems and the computation of Galois groups of difference equations. In 2012 Chen and Singer introduced discrete residues as a theoretical obstruction to summability for rational functions with respect to the shift and $q$-dilation difference operators. Since then analogous notions of discrete residues have been defined in other difference settings relevant for applications, such as for Mahler and elliptic shift difference operators. Very recently there have been some advances in making these theoretical obstructions computable in practice.