[Paper Review] A Real Nullstellensatz for Matrices of Non-Commutative Polynomials

Christopher Nelson|arXiv (Cornell University)|May 3, 2013

Advanced Topics in Algebra11 references6 citations

TL;DR

This paper establishes a noncommutative real Nullstellensatz for matrices of non-commutative polynomials in a free $\ast$-algebra $\mathbb{R}\langle x,x^\ast\rangle$, proving that a matrix polynomial vanishes on the common zero set of a finite set if and only if it lies in the smallest real left submodule containing that set. It introduces an efficient, finite-step algorithm to compute this real radical submodule, improving upon prior methods via structured Gröbner basis techniques and sum-of-squares optimization.

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 $\RR^{1 imes \ell}\axs$ 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 $\RR^{1 imes \ell}\axs$ whose zero set contains the intersection of zero sets of elements from a finite subset $S \subset \RR^{1 imes \ell}\axs$ 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 $S \subset \RR^{1 imes \ell}\axs$ 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 $\RR\axs$.

Motivation & Objective

• To extend the classical real Nullstellensatz to non-commutative settings, specifically for matrices of non-commutative polynomials in a free $\ast$-algebra.
• To define and characterize the zero set of a left submodule in $\mathbb{R}^{1\times\ell}\langle x,x^\ast\rangle$ and the notion of a real left submodule.
• To establish a noncommutative analog of the real Nullstellensatz: a matrix polynomial vanishes on the common zero set of a finite set if and only if it lies in the real radical of the submodule generated by that set.
• To develop a finite, implementable algorithm that computes the smallest real left submodule containing a given finite set of matrix polynomials.
• To improve upon existing algorithms by leveraging the rigid algebraic structure of $\mathbb{R}\langle x,x^\ast\rangle$ and using sum-of-squares techniques within a Gröbner basis framework.

Proposed method

• Introduces the concept of the zero set of a left submodule in $\mathbb{R}^{1\times\ell}\langle x,x^\ast\rangle$ and defines real left submodules via a positivity condition on sums of squares.
• Uses a recursive algorithm based on reduced left Gröbner bases to compute the real radical of a left submodule generated by a finite set of matrix polynomials.
• Employs a linear matrix inequality (LMI) formulation to test for the existence of a nontrivial sum of squares in the span of a carefully constructed monomial basis $\mathfrak{C}^{(i)}$.
• At each iteration, checks whether a sum of squares of the form $M_i^* L_i(\alpha,\beta) M_i$ exists with $L_i$ a linear pencil, using semidefinite programming techniques.
• If such a sum of squares exists, adds the corresponding polynomial vector to the generating set and recomputes a reduced left Gröbner basis; otherwise, the algorithm terminates with the real radical.
• Applies degree bounds and monomial non-divisibility conditions to ensure termination and correctness, relying on the finite-dimensionality of homogeneous components.

Experimental results

Research questions

• RQ1 Can the classical real Nullstellensatz be extended to matrices of non-commutative polynomials in a free $\ast$-algebra?
• RQ2 What is the appropriate notion of a real left submodule in the non-commutative setting, and how does it relate to zero sets?
• RQ3 Is there an algorithmic procedure to compute the smallest real left submodule containing a given finite set of matrix polynomials in $\mathbb{R}\langle x,x^\ast\rangle$?
• RQ4 How can sum-of-squares techniques be integrated into a Gröbner basis framework to compute real radicals efficiently in non-commutative algebras?
• RQ5 Does the proposed algorithm terminate in finitely many steps and correctly compute the real radical submodule?

Key findings

• The paper proves a noncommutative real Nullstellensatz: a matrix polynomial $q$ vanishes on the common zero set of a finite set $S$ if and only if $q$ lies in the smallest real left submodule containing $S$.
• The algorithm for computing the real radical submodule terminates in finitely many steps due to the finite-dimensionality of homogeneous components in $\mathbb{R}^{1\times\ell}\langle x,x^\ast\rangle$.
• Each iteration of the algorithm strictly enlarges the left module generated by the current basis, ensuring convergence.
• The algorithm maintains degree bounds: if the input polynomials have degree at most $d$, then all intermediate polynomials and monomials in the basis $\mathfrak{C}^{(i)}$ have degree at most $d-1$.
• The algorithm correctly identifies whether a nontrivial sum of squares exists via linear matrix inequality (LMI) feasibility, and if so, adds the corresponding polynomial vector to the generating set.
• When the LMI has no solution, the algorithm terminates and outputs that the current generating set is already a reduced left Gröbner basis for the real radical submodule.