The convex Positivstellensatz in a free algebra

TL;DR

This paper proves a noncommutative Positivstellensatz for convex semialgebraic sets defined by monic linear pencils, providing degree bounds for sum of squares representations of positive polynomials.

Contribution

It establishes a perfect noncommutative Nichtnegativstellensatz with explicit degree bounds, contrasting with the commutative case where such bounds are generally unavailable.

Findings

Noncommutative polynomials positive on D_L have sum of squares representations.

Degree bounds for the sum of squares are explicitly provided.

The result highlights a stark difference from the commutative setting.

Abstract

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.