A Nullstellensatz for noncommutative polynomials and factorization in free algebras
The talk concerns noncommutative polynomials f=f(x) from the perspective of free real algebraic geometry. There are several natural notions of a ``zero set'' of f. The one we discuss here is the free locus of f, Z(f), which is defined to be the union of hypersurfaces {X∈Mn(k)g:det over natural numbers n. The talk will describe a recent advance on the relation between irreducible components of Z(f) and factors of f, which was achieved using linear pencils and realizations originating in control theory.
We will start by reducing the problem to free loci Z(L) of linear pencils L, and consider a fundamental irreducibility theorem for Z(L) which is obtained with the aid of invariant theory. Next we will apply the preceding results together with P.M. Cohn's factorization theory to obtain statements about Z(f) for a noncommutative polynomial f. Finally, using Fornasini-Marchesini realizations we will describe a factorization algorithm and present applications of the Nullstellensatz to free convexity.
This is based on joint works with Meric Augat, Eric Evert, Bill Helton, Scott McCullough, and Jurij Volčič.