标题： 关于多元多项式矩阵的Smith标准型等价问题 显示英文标题

标题： On the equivalence problem of Smith forms for multivariate polynomial matrices

摘要： 本文深入研究了多变量多项式矩阵的Smith标准型的等价性问题。 一般来说，多变量（$n \geq 2$）多项式矩阵及其Smith标准型可能不等价。 然而，在某些特定条件下，我们推导出了它们等价的充要条件。 设$F\in K[x_1,\ldots,x_n]^{l\times m}$的秩为$r$，$d_r(F)\in K[x_1]$是$F$的所有$r\times r$阶子式的最大公因式，其中$K$是一个域，$x_1,\ldots,x_n$是变量且$1 \leq r \leq \min\{l,m\}$。 我们的主要发现揭示了以下结果：$F$与它的史密斯标准形等价，当且仅当$F$的所有$i\times i$阶约化子式生成$K[x_1,\ldots,x_n]$对于$i=1,\ldots,r$。 显示英文摘要

摘要： 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 $i=1,\ldots,r$.