A Real Nullstellensatz for free modules

[摘要] Let A be the algebra of all n x n matrices with entries from R[x(1), ... x(d)] and let G(1), ... ,G(m), F is an element of A. We will show that F(a)v = 0 for every a is an element of R-d and v is an element of R-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), ... ,G(m). Here a left ideal J of A is real if for every H-1, ... ,H-k is an element of A such that (H1H1)-H-T + ... + (HkHk)-H-T is an element of J + J(T) we have that H-1, ... ,H-k is an element of 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), ... ,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 R[x(1), ... ,x(d)](n). (C) 2013 Elsevier Inc. All rights reserved.

[发布日期] 2013-12-15

[关键词] Matrix polynomials;Real algebraic geometry;Real Nullstellensatz