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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2510.16825 |
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| _version_ | 1866917299490914304 |
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| author | Rowen, Louis H. Vishne, Uzi |
| author_facet | Rowen, Louis H. Vishne, Uzi |
| contents | Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th roots, for any $n \ge m'$ there are matrices $A_1,\dots,A_m$ in~$M_n(F)$ such that $f(A_1,\dots,A_m)$ is upper triangular with $n-m'$ prescribed diagonal entries.
When f is homogeneous, $f(A_1,\dots,A_m)$ is diagonal with $n-m'$ prescribed diagonal entries.
When f is multilinear, we can take $d=1$ and $m' = [\frac{m-1}{2}]$, and the upper left $(n-m')\times (n-m')$ piece of $f(A_1,\dots,A_m)$ can be taken to be $diag(β_1,\dots, β_{n-m'})$, for indeterminates $β_i$.
Furthermore, if $f$ is not a polynomial identity of $ k \times k $ matrices, then at least $ n - k $ characteristic values of $ f(A_1,\dots,A_m) $ may be taken to be algebraically independent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Makar-Limanov's problem on values of polynomials on matrices Rowen, Louis H. Vishne, Uzi Rings and Algebras Primary: 16R20, Secondary: 08B20 Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th roots, for any $n \ge m'$ there are matrices $A_1,\dots,A_m$ in~$M_n(F)$ such that $f(A_1,\dots,A_m)$ is upper triangular with $n-m'$ prescribed diagonal entries. When f is homogeneous, $f(A_1,\dots,A_m)$ is diagonal with $n-m'$ prescribed diagonal entries. When f is multilinear, we can take $d=1$ and $m' = [\frac{m-1}{2}]$, and the upper left $(n-m')\times (n-m')$ piece of $f(A_1,\dots,A_m)$ can be taken to be $diag(β_1,\dots, β_{n-m'})$, for indeterminates $β_i$. Furthermore, if $f$ is not a polynomial identity of $ k \times k $ matrices, then at least $ n - k $ characteristic values of $ f(A_1,\dots,A_m) $ may be taken to be algebraically independent. |
| title | Makar-Limanov's problem on values of polynomials on matrices |
| topic | Rings and Algebras Primary: 16R20, Secondary: 08B20 |
| url | https://arxiv.org/abs/2510.16825 |