Geometry of free loci and factorization of noncommutative polynomials

J. William Helton, Igor Klep, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


The free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces Zn(f)={X∈Mn(k)g:det⁡f(X)=0}. 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. 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.

Original languageEnglish
Pages (from-to)589-626
Number of pages38
JournalAdvances in Mathematics
StatePublished - 20 Jun 2018


  • Factorization
  • Invariant theory
  • Linear matrix inequality
  • Noncommutative polynomial
  • Singularity locus
  • Spectrahedron

ASJC Scopus subject areas

  • General Mathematics


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