Matrix evaluations of noncommutative rational functions and Waring problems
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911073280458752 |
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| author | Brešar, Matej Volčič, Jurij |
| author_facet | Brešar, Matej Volčič, Jurij |
| contents | Let $r$ be a nonconstant noncommutative rational function in $m$ variables over an algebraically closed field $K$ of characteristic 0. We show that for $n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$ distinct and nonzero eigenvalues. This result is used to study the linear and multiplicative Waring problems for matrix algebras. Concerning the linear problem, we show that for $n$ large enough, every matrix in $sl_n(K)$ can be written as $r(Y)-r(Z)$ for some $Y,Z\in M_n(K)^m$. We also discuss variations of this result for the case where $r$ is a noncommutative polynomial. Concerning the multiplicative problem, we show, among other results, that if $f$ and $g$ are nonconstant polynomials, then, for $n$ large enough, every nonscalar matrix in $GL_n(K)$ can be written as $f(Y)g(Z)$ for some $Y,Z\in M_n(K)^m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11564 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matrix evaluations of noncommutative rational functions and Waring problems Brešar, Matej Volčič, Jurij Rings and Algebras Let $r$ be a nonconstant noncommutative rational function in $m$ variables over an algebraically closed field $K$ of characteristic 0. We show that for $n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$ distinct and nonzero eigenvalues. This result is used to study the linear and multiplicative Waring problems for matrix algebras. Concerning the linear problem, we show that for $n$ large enough, every matrix in $sl_n(K)$ can be written as $r(Y)-r(Z)$ for some $Y,Z\in M_n(K)^m$. We also discuss variations of this result for the case where $r$ is a noncommutative polynomial. Concerning the multiplicative problem, we show, among other results, that if $f$ and $g$ are nonconstant polynomials, then, for $n$ large enough, every nonscalar matrix in $GL_n(K)$ can be written as $f(Y)g(Z)$ for some $Y,Z\in M_n(K)^m$. |
| title | Matrix evaluations of noncommutative rational functions and Waring problems |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2401.11564 |