Geometry of free loci and factorization of noncommutative polynomials
Helton, J. William, 1944- ; Klep, Igor, matematik ; Volčič, Jurij, 1991-
The free singularity locus of a noncommutative polynomial ▫$f$▫ is defined to be the sequence of hypersurfaces ▫$\mathscr{Z}_n(f) = \{X \in \mathrm{M}_n(\Bbbk)^g : \det f(X) = 0 \}$▫. The main ...theorem of this article shows that ▫$f$▫ is irreducible if and only if ▫$\mathscr{Z}_n(f)$▫ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Arising from this is a free singularity locus Nullstellensatz for noncommutative polynomials. Apart from 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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