Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices
Provides theoretical resolution of a long-standing conjecture in matrix theory for multivariate polynomial matrices, benefiting algebraic control theory and polynomial system solving.
The paper proves that a multivariate polynomial matrix is equivalent to its Smith normal form if and only if its reduced minors of each order generate the unit ideal, confirming a 1978 conjecture for a broad class of matrices, and extends the result to a more general setting via ring automorphisms.
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.