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Preprint
Matrix evaluations of noncommutative rational functions and Waring problems
arXiv.org
21 Jan 2024
Abstract
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 $Y,Z\in
M_n(K)^m$.
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Details
- Title
- Matrix evaluations of noncommutative rational functions and Waring problems
- Creators
- Matej BrešarJurij Volčič
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021861671104721