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Further results on equivalence of multivariate polynomial matrices

Jiancheng Guan, Jinwang Liu, Dongmei Li, Tao Wu

Abstract

This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial. We first generalized a global-local theorem of Vaserstein. Then we proved these matrices are equivalent to their Smith forms by the generalized global-local theorem.

Further results on equivalence of multivariate polynomial matrices

Abstract

This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial. We first generalized a global-local theorem of Vaserstein. Then we proved these matrices are equivalent to their Smith forms by the generalized global-local theorem.

Paper Structure

This paper contains 3 sections, 13 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Lemma 3.1

Let $B$ be a commutative ring, and $S$ be a multiplicative set in $B$. Let $\tau (x) \in {\rm GL}_n(B_S[x])$ be such that $\tau (0) = I_n$. Then there exists a matrix $\hat{\tau} (x) \in {\rm GL}_n(B[x])$ such that $\hat{\tau} (x)$ localizes to $\tau ((s / 1)x)$ (for some $s \in S$), and $\hat{\tau}

Theorems & Definitions (24)

  • Definition 2.1: Lin2001
  • Definition 2.2: Lin2001
  • Definition 2.3: Lin2001
  • Definition 2.4
  • Definition 2.5
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 14 more