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Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra
International Mathematics Research Notices ( IF 1 ) Pub Date : 2020-06-03 , DOI: 10.1093/imrn/rnaa122
J Helton 1 , Igor Klep 2 , Jurij Volčič 3
Affiliation  

This article gives a class of Nullstellensatze 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 $x^*$ one also associates its real singularity set $Z^{re}(h)=\{X: \det h(X,X^*)=0\}$. A polynomial $f$ which 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 $Z(f) \subseteq Z^{re}(h)$ is equivalent to each factor of $f$ being "stably associated" to a factor of $h$ or of $h^*$. For perspective, classical Hilbert type Nullstellensatze typically apply only to analytic polynomials $f,h $, while real Nullstellensatze 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 $V(f)=\{X: f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.

中文翻译:

自由代数的非交换多项式和 Nullstellensätze 的因式分解

本文给出了一类非交换多项式的 Nullstellensatze。非交换多项式 $f=f(x_1,\dots,x_g)$ 的奇点集是 $Z(f)=(Z_n(f))_n$,其中 $Z_n(f)=\{X \in M_n^ g: \det f(X) = 0\}.$ 这篇文章的第一个主要定理表明,$f$ 的不可约因子与 $Z_n(f)$ 的不可约分量对于每个足够大的$n$。对于 $x$ 和 $x^*$ 中的每个多项式 $h$ ,还关联了它的实奇点集 $Z^{re}(h)=\{X: \det h(X,X^*)=0\ }$。单独依赖于 $x$(没有 $x^*$ 变量)的多项式 $f$ 将被称为解析的。这里证明的主要 Nullstellensatz 如下:对于解析 $f$ 但对于可能依赖于 $x$ 和 $x^*$ 的 $h$,包含 $Z(f) \subseteq Z^{re}(h) $ 等价于 $f$ 的每个因数是“
更新日期:2020-06-03
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