Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra. (3rd June 2020)

• Record Type: Journal Article
• Authors: Helton, J; Klep, Igor; Volčič, Jurij
• Is Part Of: International mathematics research notices. Volume 2022:Number 1(2022)
• Page Start: 343
• Page End: 372
• Publication Date: 2020-06-03
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa122
• Languages: English
• ISSNs: 1073-7928

Abstract

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 $\mathscr{Z}(\, f)=(\mathscr{Z}_n(\, f))_n$, where $\mathscr{Z}_n(\, f)=\{X \in{\operatorname{M}}_{n}({\mathbb{C}})^g \colon \det f(X) = 0\}.$ The 1st main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $\mathscr{Z}_n(\, f)$ for every sufficiently large $n$ . With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $\mathscr{Z}^{{\operatorname{re}}}(h)=\{X\colon \det h(X, X^*)=0\}$ . A polynomial $f$ that depends on $x$ alone (no $x^*$ 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 $x^*$, the containment $\mathscr{Z}(\, f) \subseteq \mathscr{Z}^{{\operatorname{re}}} (h)$ is equivalent to each factor of $f$ being "stably associated" to a factor of $h$ or of $h^*$ . 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 ${\mathcal{V}}(\, f)=\{X\colon f(X, X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.