A Real Nullstellensatz for Matrices of Non-Commutative Polynomials

Abstract

This article extends the classical Real Nullstellensatz to matrices of polynomials in a free $\ast$-algebra \RR\axs with $x=(x_1, \ldots, x_n)$. This result is a generalization of a result of Cimpri\vc, Helton, McCullough, and the author. In the free left \RR\axs-module \RR^{1 \times \ell}\axs we introduce notions of the (noncommutative) zero set of a left \RR\axs-submodule and of a real left \RR\axs-submodule. We prove that every element from \RR^{1 \times \ell}\axs whose zero set contains the intersection of zero sets of elements from a finite subset S \subset \RR^{1 \times \ell}\axs belongs to the smallest real left \RR\axs-submodule containing $S$. Using this, we derive a nullstellensatz for matrices of polynomials in \RR\axs. The other main contribution of this article is an efficient, implementable algorithm which for every finite subset S \subset \RR^{1 \times \ell}\axs computes the smallest real left \RR\axs-submodule containing $S$. This algorithm terminates in a finite number of steps. By taking advantage of the rigid structure of \RR\axs, the algorithm presented here is an improvement upon the previously known algorithm for \RR\axs