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 $$\{X\in M_n(k)^g:\det f(X)=0\}$$ 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.