Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices

TL;DR

This paper establishes new theoretical criteria for when multivariate polynomial matrices are equivalent to their Smith normal form, expanding the class of matrices for which this equivalence can be determined.

Contribution

It proves Frost and Storey's 1978 conjecture for a broad class of matrices and extends the framework via polynomial ring automorphisms.

Findings

Proved the conjecture for matrices with minors generating the unit ideal.

Extended the equivalence framework to more general matrix classes.

Provided a new theoretical foundation for matrix Smith normal form analysis.

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.