New Multivariate Dimension Polynomials of Inversive Difference Field Extensions

TL;DR

This paper introduces a new class of multivariate dimension polynomials for inversive difference field extensions, providing more invariants and a method for their computation, with applications to algebraic difference equations.

Contribution

It develops a novel reduction method and characteristic set approach to define and compute multivariate dimension polynomials that capture richer invariants of difference field extensions.

Findings

New multivariate dimension polynomials describe more invariants.

Method for computing these polynomials is established.

Applications to algebraic difference equations are demonstrated.

Abstract

We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial. Then we develop the corresponding method of characteristic sets and apply it to prove the existence and obtain a method of computation of multivariate dimension polynomials of a new type that describe the transcendence degrees of intermediate fields of finitely generated inversive difference field extensions obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of $\sigma$ are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on the set of its difference generators) than standard (univariate) difference dimension polynomials. We also show how the obtained results can be applied to the equivalence problem for systems of algebraic difference equations.