Smith normal forms of bivariate polynomial matrices

TL;DR

This paper investigates conditions under which bivariate polynomial matrices are equivalent to their Smith normal forms, revealing limitations of previous assertions and extending results to broader classes using algebraic theorems.

Contribution

The paper provides a counterexample to a previous claim and proves the assertion holds for matrices with degree of det in y at most 1, extending the theory.

Findings

Counterexample for s >= 2 showing the condition is not sufficient

Proof that the assertion holds when degree of det in y is at most 1

Extension to rank-deficient and non-square matrices using Quillen-Suslin theorem

Abstract

In 1978, Frost and Storey asserted that a bivariate polynomial matrix is equivalent to its Smith normal form if and only if the reduced minors of all orders generate the unit ideal. In this paper, we first demonstrate by constructing an example that for any given positive integer s with s >= 2, there exists a square bivariate polynomial matrix M with the degree of det(M) in y equal to s, for which the condition that reduced minors of all orders generate the unit ideal is not a sufficient condition for M to be equivalent to its Smith normal form. Subsequently, we prove that for any square bivariate polynomial matrix M where the degree of det(M) in y is at most 1, Frost and Storey's assertion holds. Using the Quillen-Suslin theorem, we further extend our consideration of M to rank-deficient and non-square cases.