Desingularization of First Order Linear Difference Systems with Rational Function Coefficients

TL;DR

This paper presents algorithms to detect and remove removable singularities in first order linear difference systems with rational coefficients, improving the understanding of their meromorphic solutions.

Contribution

It introduces two algorithms for (partial) desingularization of difference systems and characterizes removable singularities via shifts of the system.

Findings

Algorithms successfully identify removable singularities.

Characterization links singularities to system shifts.

Enhanced understanding of solution meromorphic continuation.

Abstract

It is well known that for a first order system of linear difference equations with rational function coefficients, a solution that is holomorphic in some left half plane can be analytically continued to a meromorphic solution in the whole complex plane. The poles stem from the singularities of the rational function coefficients of the system. Just as for differential equations, not all of these singularities necessarily lead to poles in solutions, as they might be what is called removable. In our work, we show how to detect and remove these singularities and further study the connection between poles of solutions and removable singularities. We describe two algorithms to (partially) desingularize a given difference system and present a characterization of removable singularities in terms of shifts of the original system.