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In this paper we develop a Grobner bases theory for ideals of partialdifference polynomials with constant or non-constant coefficients. Inparticular, we introduce a criterion providing the finiteness of such baseswhen a difference ideal contains elements with suitable linear leadingmonomials. This can be explained in terms of Noetherianity of the correspondingquotient algebra. Among these Noetherian quotients we find finitely generatedpolynomial algebras where the action of suitable finite dimensional commutativealgebras and in particular finite abelian groups is defined. We obtaintherefore a consistent Grobner bases theory for ideals that possess suchsymmetries.

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