A Basis Theorem for Rings with Commuting Operators in Characteristic Zero

TL;DR

This paper establishes a basis theorem for polynomial rings with commuting generalized Hasse-Schmidt operators, unifying and extending previous differential and difference-differential basis theorems in characteristic zero.

Contribution

It introduces a new basis theorem for rings with commuting operators, generalizing Kolchin and Cohn's results in a unified framework.

Findings

Generalizes Kolchin's differential basis theorem

Extends Cohn's difference-differential basis theorem

Provides a unified basis theorem for rings with commuting operators

Abstract

Motivated by the differential basis theorem of Kolchin and the difference-differential basis theorem of Cohn, in this paper we present a basis theorem for polynomial rings equipped with commuting generalised Hasse-Schmidt operators (in the sense of Moosa and Scanlon). We recover Kolchin and Cohn's results as special cases of our main theorem.