On the equivalence problem of Smith forms for multivariate polynomial matrices

TL;DR

This paper investigates the conditions under which multivariate polynomial matrices are equivalent to their Smith forms, providing necessary and sufficient criteria based on minors, with implications for understanding polynomial matrix equivalence.

Contribution

It establishes a precise criterion for the equivalence of multivariate polynomial matrices to their Smith forms under specific conditions.

Findings

Equivalent matrices have minors generating the entire polynomial ring.

Necessary and sufficient condition for Smith form equivalence.

Characterization of polynomial matrix equivalence in multivariate case.

Abstract

This paper delves into the equivalence problem of Smith forms for multivariate polynomial matrices. Generally speaking, multivariate ($n \geq 2$) polynomial matrices and their Smith forms may not be equivalent. However, under certain specific condition, we derive the necessary and sufficient condition for their equivalence. Let $F\in K[x_1,\ldots,x_n]^{l\times m}$ be of rank $r$, $d_r(F)\in K[x_1]$ be the greatest common divisor of all the $r\times r$ minors of $F$, where $K$ is a field, $x_1,\ldots,x_n$ are variables and $1 \leq r \leq \min\{l,m\}$. Our key findings reveal the result: $F$ is equivalent to its Smith form if and only if all the $i\times i$ reduced minors of $F$ generate $K[x_1,\ldots,x_n]$ for $i=1,\ldots,r$.