The convex Positivstellensatz in a free algebra

Abstract Given a monic linear pencil L in g variables, let P L = ( P L ( n ) ) n ∈ N where P L ( n ) : = { X ∈ S n g ∣ L ( X ) ⪰ 0 } , and S n g is the set of g -tuples of symmetric n × n matrices. Because L is a monic linear pencil, each P L ( n ) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form P L . The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative matrix-valued polynomial p is positive semidefinite on P L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s ∗ s + ∑ j finite f j ∗ L f j , where s , f j are matrices of noncommutative polynomials of degree no greater than deg ( p ) 2 . 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.