Geometry of Free Loci and Factorization of Noncommutative Polynomials

Öffnen
Datum
2017-10-02MFO Scientific Program
Research in Pairs 2017Serie
Oberwolfach Preprints;2017,23Autor
Helton, J. William
Klep, Igor
Volčič, Jurij
Helton, J. William
Metadata
Zur LanganzeigeOWP-2017-23
Zusammenfassung
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.