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arXiv · 1708.05378

Geometry of free loci and factorization of noncommutative polynomials

J. William Helton·Igor Klep·Jurij Volčič
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Abstract

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ 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.

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J. William Helton, Igor Klep, Jurij Volčič. 2017-08-17. Geometry of free loci and factorization of noncommutative polynomials. https://doi.org/10.1016/j.aim.2018.04.007

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