Groebner Bases Applied to Systems of Linear Difference Equations

TL;DR

This paper introduces an algorithm and Maple implementation for transforming systems of linear difference equations into Groebner basis form, aiding in scientific computing and physics applications.

Contribution

It presents a novel algorithm and implementation for Groebner basis transformation of linear difference systems, applicable to PDE discretization and Feynman integral reduction.

Findings

Algorithm successfully transforms difference systems into Groebner basis form

Implementation in Maple facilitates automatic generation of difference schemes

Applicable to reduction of Feynman integrals and PDE discretization

Abstract

In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high energy physics as recurrence relations for multiloop Feynman integrals. The most universal algorithmic tool for investigation of linear difference systems is based on their transformation into an equivalent Groebner basis form. We present an algorithm for this transformation implemented in Maple. The algorithm and its implementation can be applied to automatic generation of difference schemes for linear partial differential equations and to reduction of Feynman integrals. Some illustrative examples are given.