Mathematics – Rings and Algebras
Scientific paper
2011-02-23
Mathematics
Rings and Algebras
22 pages
Scientific paper
Given a monic linear pencil L in g variables let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form D_L. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative polynomial p is positive semidefinite on D_L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s^* s + \sum_j f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s, f_j and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.
Helton John William
Klep Igor
McCullough Scott
No associations
LandOfFree
The convex Positivstellensatz in a free algebra does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The convex Positivstellensatz in a free algebra, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The convex Positivstellensatz in a free algebra will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-85867