Computation of Difference Groebner Bases

TL;DR

This paper introduces an algorithm for computing difference Groebner bases using Janet-like division, implemented in Maple, with applications in differential equations and Feynman integrals.

Contribution

It presents a new involutive algorithm for difference Groebner bases, including implementation details and applications to linear and nonlinear difference polynomial ideals.

Findings

Implemented in Maple as the LDA package

Successfully generates finite difference approximations to PDEs

Reduces Feynman integrals using the new algorithm

Abstract

To compute difference Groebner bases of ideals generated by linear polynomials we adopt to difference polynomial rings the involutive algorithm based on Janet-like division. The algorithm has been implemented in Maple in the form of the package LDA (Linear Difference Algebra) and we describe the main features of the package. Its applications are illustrated by generation of finite difference approximations to linear partial differential equations and by reduction of Feynman integrals. We also present the algorithm for an ideal generated by a finite set of nonlinear difference polynomials. If the algorithm terminates, then it constructs a Groebner basis of the ideal.