Journal article
FREE BERTINI'S THEOREM AND APPLICATIONS
Proceedings of the American Mathematical Society, v 148(9), pp 3661-3671
01 Sep 2020
Abstract
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 circle 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 x n matrices: lambda I > 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.
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Details
- Title
- FREE BERTINI'S THEOREM AND APPLICATIONS
- Creators
- Jurij Volcic - Texas A&M University
- Publication Details
- Proceedings of the American Mathematical Society, v 148(9), pp 3661-3671
- Publisher
- Amer Mathematical Soc
- Number of pages
- 11
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000550683900001
- Scopus ID
- 2-s2.0-85090542439
- Other Identifier
- 991021861673804721
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- Web of Science research areas
- Mathematics
- Mathematics, Applied