On the equivalence problem of Smith forms for multivariate polynomial matrices
Dong Lu, Dingkang Wang, Fanghui Xiao, Xiaopeng Zheng
TL;DR
The paper addresses when a multivariate polynomial matrix $F \in K[x_1,\ldots,x_n]^{l\times m}$ of rank $r$ is equivalent to its Smith form, focusing on the case $\det(F)\in K[x_1]$ and establishing a sharp criterion: $F$ is equivalent to its Smith form if and only if every $i\times i$ reduced minor of $F$ generates the whole ring, i.e., $J_i(F)=K[x_1,\ldots,x_n]$ for $i=1,\ldots,r$. The authors develop a localization-based approach and leverage Suslin stability and the Quillen–Suslin theorem to handle non-square and non-full-rank cases, generalizing prior bivariate results. The main theorem provides a practical, algebraic condition via reduced minors, with a corollary that extends to the general setting using standard reductions, potentially impacting both theory and applications in multidimensional system theory and symbolics. Overall, the work broadens the scope of Frost–Storey type criteria to multivariate polynomial matrices and offers a constructive path to Smith form equivalence via localization and projective-module techniques.
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 $i=1,\ldots,r$.