Structure and arithmetic of multivariate Ore extensions

TL;DR

This paper explores the structure of multivariate Ore extensions, introduces pseudo multilinear transformations related to modules over these extensions, and provides a product formula to analyze roots of polynomials within this framework.

Contribution

It introduces pseudo multilinear transformations and a product formula for multivariate Ore extensions, advancing understanding of their structure and roots.

Findings

Pseudo multilinear transformations are linked to modules over Ore extensions.

A general product formula for polynomials in Ore extensions is established.

Structure of roots of polynomials in multivariate Ore extensions is clarified.

Abstract

We give the basic structure of the multivariable Ore extensions $S=A[\underline{t} ; \sigma, \underline{\delta}]$ introduced in the work of Mart\'inez-Pe\~nas and Kschischang. The Pseudo multilinear transformations (PMT's) are introduced and correspond to modules over $S$. These maps are strongly connected to the evaluation of polynomials in $S$. A general product formula is obtained. PMT's help to put some structure on the set of roots of a polynomial $f(t) \in S$.