Separating inequalities for nonnegative polynomials that are not sums of squares

Sadik Iliman (1) , Timo de Wolff (2)

1 Goethe University Frankfurt
2 Universität des Saarlandes [Saarbrücken]

Ternary sextics and quaternary quartics are the smallest cases where there exist nonnegative polynomials that are not sums of squares (SOS). A complete classification of the difference between these cones was given by G. Blekherman via analyzing the extreme rays of the corresponding dual cones. However, an exact computational approach in order to build separating extreme rays for nonnegative polynomials that are not sums of squares is a widely open problem. We provide a method substantially simplifying this computation for certain classes of polynomials on the boundary of the PSD cones. In particular, our method yields separating extreme rays for every nonnegative ternary sextic with at least seven zeros, which proves a slight variation of a conjecture by Blekherman for many instances. As an application, we compute rational certificates for some prominent polynomials.

Domaines: Géométrie algébrique [math.AG]

Sadik Iliman, Timo de Wolff. Separating inequalities for nonnegative polynomials that are not sums of squares. MEGA'2013 (Special Issue), Jun 2013, Frankfurt am Main, Allemagne. ⟨hal-01022892⟩