Noetherian quotients of the algebra of partial difference polynomials and Grobner bases of symmetric ideals

TL;DR

This paper develops a Grobner bases theory for partial difference polynomial ideals, establishing conditions for finite bases and exploring Noetherian quotient algebras with symmetries like finite groups.

Contribution

It introduces a criterion for finiteness of Grobner bases in difference ideals and characterizes Noetherian quotients with symmetries, expanding algebraic tools for symmetric ideals.

Findings

Finiteness criterion for Grobner bases in difference ideals

Identification of Noetherian quotient algebras with symmetries

Development of a Grobner bases theory for symmetric ideals

Abstract

In this paper we develop a Grobner bases theory for ideals of partial difference polynomials with constant or non-constant coefficients. In particular, we introduce a criterion providing the finiteness of such bases when a difference ideal contains elements with suitable linear leading monomials. This can be explained in terms of Noetherianity of the corresponding quotient algebra. Among these Noetherian quotients we find finitely generated polynomial algebras where the action of suitable finite dimensional commutative algebras and in particular finite abelian groups is defined. We obtain therefore a consistent Grobner bases theory for ideals that possess such symmetries.