A Real Nullstellensatz for Free Modules

Abstract

Let $A$ be the algebra of all $n\times n$ matrices with entries from $\RR[x_1,\ldots,x_d]$ and let $G_1,\dots ,G_m,F\in A$. We will show that $F(a)v=0$ for every $a \in \RR^d$ and $v \in \RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\dots ,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\dots ,H_k\in A$ such that $H_1^T{H}_1+\dots +H_k^T{H}_k\in J+J^T$ we have that $H_1,\dots ,H_k\in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $n$ that it holds when $G_1,\dots ,G_m,F$ have zeros everywhere except in the first row. This auxiliary result can be formulated as a Real Nullstellensatz for the free module $\RR[x_1,\ldots,x_d]^n$.

v1 7 pages. v2 9 pages: revised abstract, extended introduction and references. To appear in J. Algebra

References in corpus (5)

• An exact duality theory for semidefinite programming based on sums of squares
• Moment problems for operator polynomials
• On real one-sided ideals in a free algebra
• A Real Nullstellensatz for Matrices of Non-Commutative Polynomials
• Algebras of Fractions and Strict Positivstellensätze for *-Algebras

Cited by in corpus (3)

• A Real Nullstellensatz for Matrices of Non-Commutative Polynomials
• Finsler's Lemma for Matrix Polynomials
• A nullstellensatz for linear partial differential equations with polynomial coefficients