2013/05/03 by Christopher Nelson, Nelson, Christopher S.
Computer Science · Engineering · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Operator Algebras (math.OA) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1305.0799
openalex publication_date 2013/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article extends the classical Real Nullstellensatz to matrices of polynomials in a free ∗-algebra \RR\axs with x=(x1, …, xn). This result is a generalization of a result of Cimpri\vc, Helton, McCullough, and the author. In the free left \RR\axs-module \RR1 × ℓ\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 \RR1 × ℓ\axs whose zero set contains the intersection of zero sets of elements from a finite subset S ⊂ \RR1 × ℓ\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 ⊂ \RR1 × ℓ\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.