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 $i=1,\ldots,r$.

翻译：本文深入探讨了多元多项式矩阵Smith形式的等价性问题。一般而言，多元（$n \geq 2$）多项式矩阵与其Smith形式可能并不等价。然而，在特定条件下，我们推导出了二者等价的充要条件。设 $F\in K[x_1,\ldots,x_n]^{l\times m}$ 的秩为 $r$，$d_r(F)\in K[x_1]$ 为 $F$ 的所有 $r\times r$ 子式的最大公因式，其中 $K$ 为域，$x_1,\ldots,x_n$ 为变量，且 $1 \leq r \leq \min\{l,m\}$。我们的核心研究结果表明：$F$ 与其Smith形式等价当且仅当对 $i=1,\ldots,r$，$F$ 的所有 $i\times i$ 约化子式生成整环 $K[x_1,\ldots,x_n]$。