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  • Free Bertini's theorem and applications
    Volčič, Jurij, 1991-
    The simplest version of Bertini's irreducibility theorem states that the generic fiber of a noncomposite polynomial function is an irreducible hypersurface. The main result of this paper is its ... analog for a free algebra: if ▫$f$▫ is a noncommutative polynomial such that▫ $f - \lambda$▫ factors for infinitely many scalars ▫$\lambda$▫, then there exist a noncommutative polynomial ▫$h$▫ and a nonconstant univariate polynomial ▫$p$▫ such that ▫$f = p\circ h$▫. Two applications of free Bertini's theorem for matrix evaluations of noncommutative polynomials are given. An eigenlevel set of ▫$f$▫ is the set of all matrix tuples ▫$X$▫ where ▫$f(X)$▫ attains some given eigenvalue. It is shown that eigenlevel sets of ▫$f$▫ and ▫$g$▫ coincide if and only if ▫$fa = ag$▫ for some nonzero noncommutative polynomial ▫$a$▫. The second application pertains to quasiconvexity and describes polynomials ▫$f$▫ such that the connected component of ▫$\{X$▫ tuple of symmetric ▫$n\times n$▫ matrices: ▫$\lambda I\succ f(X)\}$▫ about the origin is convex for all natural ▫$n$▫ and ▫$\lambda > 0$▫. It is shown that such a polynomial is either everywhere negative semidefinite or the composition of a univariate and a convex quadratic polynomial.
    Vir: Proceedings of the American Mathematical Society. - ISSN 0002-9939 (Vol. 148, no. 9, Sep. 2020, str. 3661-3671)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2020
    Jezik - angleški
    COBISS.SI-ID - 48667139

    Povezava(-e):

    https://doi.org/10.1090/proc/15071
    DOI

Vnos na polico