Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals. The case of one translation

TL;DR

This paper develops a method to compute bivariate Kolchin-type dimension polynomials for non-reflexive difference-differential ideals, enhancing understanding of their structure and applications to algebraic difference-differential equations.

Contribution

It introduces a new approach using characteristic sets to compute dimension polynomials for non-reflexive ideals, providing a novel proof and computational method.

Findings

Established existence of bivariate Kolchin-type dimension polynomials.

Provided a method to compute the dimension polynomial of non-reflexive prime difference ideals.

Linked the reflexive closure of difference polynomial ideals to inverse images under translation.

Abstract

We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate Kolchin-type dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. In particular, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. As a consequence, it is shown that the reflexive closure of a prime difference polynomial ideal is the inverse image of this ideal under a power of the basic translation. We also discuss applications of our results to the analysis of systems of algebraic difference-differential equations.