Degree lower bounds in Positivstellensatz via Tropical Techniques

Positivstellensatz is an important result in real algebraic geometry which guarantees that a polynomial strictly positive on a compact semialgebraic set has a representation as an "obviously nonnegative" polynomial via a sum of squares certificate. This certificate is computationally tractable via semidefinite programming (SDP), but the size of the SDP is dependent on the certificate's degree. It is important to understand how the degree of the certificate depends on the minimum of the positive polynomial on the semialgebraic set. There has been a recent flurry of activity providing upper bounds on the degree, but comparatively little is known about lower bounds. I will present some ideas on how to use tropical techniques to prove lower bounds. This is work in progress, and I hope the audience will help it progress even further.