Title: A Perfect Positivstellensatz in a Free Algebra

Abstract: A positivstellensatz is an algebraic certificate for a given polynomial p to have a specific positivity property. Such theorems date back in some form for over one hundred years for conventional (commutative) polynomials. For example, given polynomials p, q, and r, one might ask

(Q) Does q(x) ≥ 0 and r(x) = 0 imply p(x) ≥ 0?

In this talk I will focus on free noncommutative polynomials p, q, and r and substitute matrices for the variables xj. In the case that the positivity domain D ∶={X | p(X) ⪰ 0} is convex, there exists a “perfect positivstellensatz”, which gives an elegant answer to (Q) with strict degree bounds. This non-commutative positivstellensatz has a much cleaner statement than analogous commutative results. In addition, from this result follow a few corollaries, including a non-commutative nullstellensatz as well as results relating to convex optimization.