On the length of generating sets with conditions on minimal polynomial
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913818966228992 |
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| author | Wang, Chengjie |
| author_facet | Wang, Chengjie |
| contents | Linear upper bounds may be derived by imposing specific structural conditions on a generating set, such as additional constraints on ranks, eigenvalues, or the degree of the minimal polynomial of the generating matrices. This paper establishes a linear upper bound of \(3n-5\) for generating sets that contain a matrix whose minimal polynomial has a degree exceeding \(\frac{n}{2}\), where \(n\) denotes the order of the matrix. Compared to the bound provided in \cite[Theorem 3.1]{r2}, this result reduces the constraints on the Jordan canonical forms. Additionally, it is demonstrated that the bound \(\frac{7n}{2}-4\) holds when the generating set contains a matrix with a minimal polynomial of degree \(t\) satisfying \(2t\le n\le 3t-1\). The primary enhancements consist of quantitative bounds and reduced reliance on Jordan form structural constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the length of generating sets with conditions on minimal polynomial Wang, Chengjie Rings and Algebras Linear upper bounds may be derived by imposing specific structural conditions on a generating set, such as additional constraints on ranks, eigenvalues, or the degree of the minimal polynomial of the generating matrices. This paper establishes a linear upper bound of \(3n-5\) for generating sets that contain a matrix whose minimal polynomial has a degree exceeding \(\frac{n}{2}\), where \(n\) denotes the order of the matrix. Compared to the bound provided in \cite[Theorem 3.1]{r2}, this result reduces the constraints on the Jordan canonical forms. Additionally, it is demonstrated that the bound \(\frac{7n}{2}-4\) holds when the generating set contains a matrix with a minimal polynomial of degree \(t\) satisfying \(2t\le n\le 3t-1\). The primary enhancements consist of quantitative bounds and reduced reliance on Jordan form structural constraints. |
| title | On the length of generating sets with conditions on minimal polynomial |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2504.17348 |