Simple polynomial equations over (mxm)-matrices
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908442493452288 |
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| author | Chatyrko, Vitalij A. Karassev, Alexandre |
| author_facet | Chatyrko, Vitalij A. Karassev, Alexandre |
| contents | Let $m$ be any integer $\geq 3$. We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(m \times m)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb R$ for each $i$ and $n \geq 1$. We discuss its solution set $S$ supplied with the natural Euclidean topology. In particular, we describe the solution set $S$ for $m=3$ and calculate its dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07085 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple polynomial equations over (mxm)-matrices Chatyrko, Vitalij A. Karassev, Alexandre Rings and Algebras General Topology Primary: 15A24, 54F45, Secondary: 15B30 Let $m$ be any integer $\geq 3$. We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(m \times m)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb R$ for each $i$ and $n \geq 1$. We discuss its solution set $S$ supplied with the natural Euclidean topology. In particular, we describe the solution set $S$ for $m=3$ and calculate its dimension. |
| title | Simple polynomial equations over (mxm)-matrices |
| topic | Rings and Algebras General Topology Primary: 15A24, 54F45, Secondary: 15B30 |
| url | https://arxiv.org/abs/2507.07085 |