The singularity set of a non-commutative polynomial f=f(x1, . . . ,xd) is the graded set Z(f)=(Zn(f))n, where Zn(f)={ X ∈ Mnd : det f(X) = 0 }. Two main results will be presented. First, irreducible factors of f are shown to be in a natural bijective correspondence with irreducible components of Zn(f) for every sufficiently large n. In particular, f is irreducible if and only if Zn(f) is eventually irreducible. Second, we give Nullstellensätze for non-commutative polynomials. For instance, given two non-commutative polynomials f1, f2, we have Z(f1) ⊆ Z(f2) if and only if each irreducible factor of f1 is (up to stable associativity) an irreducible factor of f2. Along the way an algorithm for factorization of non-commutative polynomials will be presented.
The talk is based on joint work with Jurij Volcic and Bill Helton.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
