On the equivalence problem of Smith forms for multivariate polynomial matrices
<|reference_start|>On the equivalence problem of Smith forms for multivariate polynomial matrices: 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 $i=1,\ldots,r$.<|reference_end|>
@article{lu2024on,
    title={On the equivalence problem of Smith forms for multivariate polynomial
  matrices},
    author={Dong Lu, Dingkang Wang, Fanghui Xiao, Xiaopeng Zheng},
    journal={arXiv preprint arXiv:2407.06649},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.06649},
    primaryClass={cs.SC math.AC}
}