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  • Matrix evaluations of noncommutative rational functions and Waring problems
    Brešar, Matej ; Volčič, Jurij, 1991-
    Let ▫$r$▫ be a nonconstant noncommutative rational function in ▫$m$▫ variables over an algebraically closed field ▫${\mathbb K}$▫ of characteristic ▫$0$▫. We show that for ▫$n$▫ large enough, there ... exists an ▫$X\in {\rm M}_n({\mathbb K})^m$▫ 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 ▫$\mathfrak{sl}_n({\mathbb K})$▫ can be written as ▫$r(Y)-r(Z)$▫ for some ▫$Y,Z\in {\rm M}_n({\mathbb K})^m$▫. 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 ▫${\rm GL}_n({\mathbb K})$▫ can be written as ▫$f(Y) \cdot g(Z)$▫ for some ▫$Y,Z\in {\rm M}_n({\mathbb K})^m$▫.
    Vir: Selecta mathematica. New series. - ISSN 1022-1824 (Vol. 31, iss. 5, [article no.] 97, Nov. 2025, 16 str.)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2025
    Jezik - angleški
    COBISS.SI-ID - 254206211

    Povezava(-e):

    https://link.springer.com/article/10.1007/s00029-025-01098-7
    DOI

vir: Selecta mathematica. New series. - ISSN 1022-1824 (Vol. 31, iss. 5, [article no.] 97, Nov. 2025, 16 str.)
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