Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices
Commutative Algebra
2026-05-12 v1 Symbolic Computation
Abstract
This paper investigates the Smith normal form equivalence problem for multivariate polynomial matrices. Using methods from matrix theory and polynomial ideal theory, we prove that Frost and Storey's 1978 conjecture holds for a broad class of matrices: such a matrix is equivalent to its Smith normal form if and only if its reduced minors of each order generate the unit ideal. Moreover, by extending the original matrix class via automorphisms of the polynomial ring, we show that our framework applies in a substantially more general setting.
Cite
@article{arxiv.2605.09286,
title = {Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices},
author = {Dong Lu and Yuanyuan Ruan and Dingkang Wang and Fanghui Xiao},
journal= {arXiv preprint arXiv:2605.09286},
year = {2026}
}