The geometry of SOS-multipliers on varieties

A homogeneous polynomial F admits an SOS-multiplier certificate on a real projective variety X if there exist sums of squares g and s such that Fg=s in the homogeneous coordinate ring of X. Such an expression certifies the nonnegativity of F and is thus of considerable theoretical and practical importance. In this talk I will present ongoing work with G. Blekherman, R. Sinn and G.G. Smith on the geometry of such nonnegativity certificates. Our main results are effective bounds on the degrees of possible g's on algebraic curves and surfaces which depend on classical geometric invariants of the varieties. These bounds are, in some cases, provably optimal.