Hilbert-type dimension polynomials of intermediate difference-differential field extensions

TL;DR

This paper introduces Hilbert-type dimension polynomials for intermediate difference-differential field extensions, linking algebraic properties with numerical polynomials and extending to multidimensional filtrations.

Contribution

It develops a new polynomial invariant for difference-differential extensions and generalizes the concept to multidimensional filtrations, enhancing the understanding of their algebraic structure.

Findings

Transcendence degrees are expressed by a numerical polynomial.

Properties of the polynomials are established and linked to Krull-type dimension.

Results are extended to multidimensional filtrations.

Abstract

Let $K$ be an inversive difference-differential field and $L$ a (not necessarily inversive) finitely generated difference-differential field extension of $K$. We consider the natural filtration of the extension $L/K$ associated with a finite system $\eta$ of its difference-differential generators and prove that for any intermediate difference-differential field $F$, the transcendence degrees of the components of the induced filtration of $F$ are expressed by a certain numerical polynomial $\chi_{K, F,\eta}(t)$. This polynomial is closely connected with the dimension Hilbert-type polynomial of a submodule of the module of K\"ahler differentials $\Omega_{L^{\ast}|K}$ where $L^{\ast}$ is the inversive closure of $L$. We prove some properties of polynomials $\chi_{K, F,\eta}(t)$ and use them for the study of the Krull-type dimension of the extension $L/K$. In the last part of the paper, we present a generalization of the obtained results to multidimensional filtrations of $L/K$ associated with partitions of the sets of basic derivations and translations.