Smith normal forms of bivariate polynomial matrices
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915414654582784 |
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| author | Lu, Dong Wang, Dingkang Xiao, Fanghui Zheng, Xiaopeng |
| author_facet | Lu, Dong Wang, Dingkang Xiao, Fanghui Zheng, Xiaopeng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20889 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smith normal forms of bivariate polynomial matrices Lu, Dong Wang, Dingkang Xiao, Fanghui Zheng, Xiaopeng Symbolic Computation Rings and Algebras 68W30, 15A24, 13P10 I.1.1; I.1.2 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. |
| title | Smith normal forms of bivariate polynomial matrices |
| topic | Symbolic Computation Rings and Algebras 68W30, 15A24, 13P10 I.1.1; I.1.2 |
| url | https://arxiv.org/abs/2507.20889 |