Smith normal forms of bivariate polynomial matrices

2025-07-28

AI Summary

This paper investigates necessary and sufficient conditions for a bivariate polynomial matrix to be equivalent to its Smith normal form, focusing on the completeness of the Frost–Storey conjecture—which asserts that a matrix is diagonalizable (i.e., equivalent to a diagonal matrix) if and only if all its reduced minors generate the unit ideal. We first construct explicit counterexamples with $deg_y det M geq 2$, thereby disproving the sufficiency of the Frost–Storey condition in general. We then prove that the conjecture holds when $deg_y det M leq 1$. Leveraging the Quillen–Suslin theorem, we extend this result to rank-deficient and non-square matrices. Our work unifies the applicability boundaries of classical equivalence criteria and broadens the scope of structural theory for polynomial matrices.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Identify insufficiency of Frost-Storey condition for Smith normal forms
• Prove Frost-Storey assertion holds for det(M) degree ≤1 in y
• Extend analysis to rank-deficient and non-square matrices using Quillen-Suslin

Innovation

• Constructed counterexample for Smith normal form
• Proved assertion for degree ≤1 in y
• Extended analysis using Quillen-Suslin theorem