Multivariate Difference-Differential Dimension Polynomials and New Invariants of Difference-Differential Field Extensions

TL;DR

This paper develops a new method for computing multivariate difference-differential dimension polynomials and introduces invariants for difference-differential field extensions, aiding in solving algebraic systems.

Contribution

It presents a novel characteristic set approach for difference-differential polynomials and identifies new invariants for field extensions.

Findings

Method for computing multivariate dimension polynomials

Introduction of new invariants of difference-differential extensions

Application to algebraic difference-differential equations

Abstract

In this paper we introduce a method of characteristic sets with respect to several term orderings for difference-differential polynomials. Using this technique, we obtain a method of computation of multivariate dimension polynomials of finitely generated difference-differential field extensions. Furthermore, we find new invariants of such extensions and show how the computation of multivariate difference-differential polynomials is applied to the equivalence problem for systems of algebraic difference-differential equations.