Analysis Seminar: "Positive polynomials"

Organizer: John McCarthy

Abstract:  The understanding positivity sets for polynomials mirrors the understanding of zero sets for polynomials. For example, given that some set of polynomials vanish on a set, we can use Hilbert's Nullstellensatz (along with effective methods such as Groebner bases which were developed much later) to understand if some other polynomial also vanishes on that same set. Similarly, given that some set of polynomials are positive on a given set, we can use Putinar’s Positivstellensatz  to know if some other polynomial is also positive on that same set. Following Putinar’s methods, in the early 2000s, Helton and his collaborators began work on understanding systems of matrix inequalities from engineering. We will survey known results on positive polynomials and systems of inequalities and give a simplified proof of Helton’s result from the paper 2002 "Positive nonommutative polynomials are  sums of squares."