**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 *x _{j}*. 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.