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

2026-05-09

AI Summary

This study addresses the classical problem of whether multivariate polynomial matrices are equivalent to their Smith normal forms, with a focus on verifying the conjecture proposed by Frost and Storey (1978). By integrating techniques from matrix theory, polynomial ideal theory, and automorphisms of polynomial rings, the authors analyze the structural properties of ideals generated by reduced minors of all orders. They establish a necessary and sufficient condition for such equivalence: the ideals generated by the reduced minors of every order must be the unit ideal. This result not only confirms the Frost–Storey conjecture for a broad class of multivariate polynomial matrices but also extends its validity—via ring automorphisms—to a significantly wider family of matrices, thereby substantially broadening the scope of existing theory.

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.

Problem

• Smith normal form
• multivariate polynomial matrices
• matrix equivalence
• reduced minors
• polynomial ideal

Innovation

• Smith normal form
• multivariate polynomial matrices
• matrix equivalence
• polynomial ideal theory
• ring automorphism