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Title: Geometry of free loci and factorization of noncommutative polynomials
Authors: Helton, J. WilliamKlep, IgorVolčič, Jurij
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Issue Date: 2017
Published in: Oberwolfach Preprints (OWP), Volume 2017-23, ISSN 1864-7596
Publisher: Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach
Abstract: The free singularity locus of a noncommutative polynomial f is defined to be the sequence Zn(f)={X∈Mgn:detf(X)=0} of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if Zn(f) is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.
Keywords: Noncommutative polynomial; factorization; singularity locus; linear matrix inequality; spectrahedron; real algebraic geometry; realization; free algebra; invariant theory
DDC: 510
License: CC BY-NC-SA 4.0 Unported
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