Multivariate Difference-Differential Dimension Polynomials

TL;DR

This paper extends algebraic methods to compute multivariate difference-differential dimension polynomials, providing a theoretical framework and computational techniques for analyzing finitely generated difference-differential field extensions.

Contribution

It generalizes the Ritt-Kolchin characteristic set method and Gr"obner basis techniques to multivariate difference-differential polynomials, offering new tools for their computation and interpretation.

Findings

Established existence of multivariate difference-differential dimension polynomials.

Developed algorithms for their computation.

Provided an interpretation related to PDE system strength.

Abstract

In this paper we generalize the Ritt-Kolchin method of characteristic sets and the classical Gr\"obner basis technique to prove the existence and obtain methods of computation of multivariate difference-differential dimension polynomials associated with a finitely generated difference-differential field extension. We also give an interpretation of such polynomials in the spirit of the A. Einstein's concept of strength of a system of PDEs and determine their invariants, that is, characteristics of a finitely generated difference-differential field extension carried by every its dimension polynomial.