Factorization of noncommutative polynomials and Nullstellensätze for the free algebra

Abstract

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots ,x_g)$ is $Z(f)=(Z_n(f){)}_n$, where $Z_n(f)=\{X\in M_n^g:\det f(X)=0\}.$ The first main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $Z_n(f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^{\ast }$ one also associates its real singularity set $Z^{re}(h)=\{X:\det h(X,X^{\ast })=0\}$. A polynomial $f$ which depends on $x$ alone (no $x^{\ast }$ variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic $f$ but for $h$ dependent on possibly both $x$ and $x^{\ast }$, the containment $Z(f)\subseteq Z^{re}(h)$ is equivalent to each factor of $f$ being "stably associated" to a factor of $h$ or of $h^{\ast }$. For perspective, classical Hilbert type Nullstellensätze typically apply only to analytic polynomials $f,h$, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above "algebraic certificate" does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018) 589-626) obtained such a theorem for special classes of analytic polynomials $f$ and $h$. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros $V(f)=\{X:f(X,X^{\ast })=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.