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 ≥ 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∈ K[x₁,…,xₙ]l× m be of rank r, dᵣ(F)∈ K[x₁] be the greatest common divisor of all the r× r minors of F, where K is a field, x₁,…,xₙ are variables and 1 ≤ r ≤ min,m\. Our key findings reveal the result: F is equivalent to its Smith form if and only if all the i× i reduced minors of F generate K[x₁,…,xₙ] for i=1,…,r.