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Structure and arithmetic of multivariate Ore extensions

André Leroy, Huda Merdach

TL;DR

The paper develops the structure theory of multivariable Ore extensions $S= A[\\underline{t}; \\sigma,\\underline{\\delta}]$, introducing pseudo-multilinear transformations (PMT) that realize $S$-actions on left $A$-modules and connect to the evaluation of polynomials in $S$. It establishes a general product formula and uses PMTs to characterize the root sets $V(f)$ via conjugacy classes, while also analyzing the center, semi-invariant polynomials, and centralizers. The results extend univariate skew-polynomial theory to the multivariable setting and provide a framework for studying module morphisms, root decompositions, and centralizer structures, with potential connections to noncommutative algebra and coding theory. The approach unifies evaluation, conjugation, and module-theoretic perspectives, offering tools to analyze roots and morphisms in multivariate noncommutative polynomial rings.

Abstract

We give the basic structure of the multivariable Ore extensions $S=A[\underline{t} ; σ, \underlineδ]$ introduced in the work of Martínez-Peñas 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 $f(t) \in S$.

Structure and arithmetic of multivariate Ore extensions

TL;DR

The paper develops the structure theory of multivariable Ore extensions , introducing pseudo-multilinear transformations (PMT) that realize -actions on left -modules and connect to the evaluation of polynomials in . It establishes a general product formula and uses PMTs to characterize the root sets via conjugacy classes, while also analyzing the center, semi-invariant polynomials, and centralizers. The results extend univariate skew-polynomial theory to the multivariable setting and provide a framework for studying module morphisms, root decompositions, and centralizer structures, with potential connections to noncommutative algebra and coding theory. The approach unifies evaluation, conjugation, and module-theoretic perspectives, offering tools to analyze roots and morphisms in multivariate noncommutative polynomial rings.

Abstract

We give the basic structure of the multivariable Ore extensions introduced in the work of Martínez-Peñas and Kschischang. The Pseudo multilinear transformations (PMT's) are introduced and correspond to modules over . These maps are strongly connected to the evaluation of polynomials in . A general product formula is obtained. PMT's help to put some structure on the set of roots of a polynomial .
Paper Structure (5 sections, 15 theorems, 17 equations)

This paper contains 5 sections, 15 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Structure of multivariate Ore extensions
  3. Center of S and Semi-invariant polynomials
  4. Evaluation and $(\sigma,\underline{\delta})$-conjugation
  5. Centralizers and roots

Key Result

Proposition 2.2

(1) The associativity of the ring $S$ leads to the following rule for the maps $\delta_1,\dots,\delta_n$: In a compact form, this can be written as $\underline{\delta}(ab)=\sigma(a)\underline{\delta}(b)+\underline{\delta}(a)b.$ The sequence of maps $\delta_{\underline{a}}$ will be called a $\sigma$-derivation. (2) The fact that $\sigma$ and $\underline{\delta}$ satisfy the above properties can al

Theorems & Definitions (33)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.4
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • proof
  • Theorem 2.8
  • proof
  • Proposition 3.1
  • ...and 23 more