标题： 二元多项式矩阵的Smith标准型 显示英文标题

标题： Smith normal forms of bivariate polynomial matrices

摘要： 1978年，Frost和Storey断言，当且仅当所有阶数的约化子式生成单位理想时，二元多项式矩阵与其史密斯标准形等价。 在本文中，我们首先通过构造一个例子来证明，对于任何给定的正整数s且s >= 2，存在一个平方的二元多项式矩阵M，其det(M)在y中的次数等于s，对于该矩阵，所有阶数的约化子式生成单位理想的条件并不是M与其史密斯标准形等价的充分条件。 随后，我们证明了对于任何平方的二元多项式矩阵M，其det(M)在y中的次数最多为1时，Frost和Storey的断言成立。 利用Quillen-Suslin定理，我们进一步将对M的考虑扩展到秩亏缺和非方阵的情况。 显示英文摘要

摘要： 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.