Geometry of free loci and factorization of noncommutative polynomials

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.

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https://doi.org/10.34657/2426
https://oa.tib.eu/renate/handle/123456789/2613

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Mathematik
Oberwolfach Preprints (OWP)

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