Georgia Institute of Technology College of Sciences

If we can write a (homogeneous) polynomial as a sum of squares(SOS), the polynomial is guaranteed to be a non-negative polynomial. However, every non-negative forms does not have to be written as sums of squares in general. This implies that set of sums of square is strictly contained in set of non-negative forms in general. We want to discuss about one way to describe the gaps between the two sets. Since the sets have cone structures, we can consider dual cones of each cones. In particular, the description of dual cone of non-negative polynomials is simple: convex hull of point evaluations. Therefore, we are interested in positive semi-definite quadratic forms that is not point evaluations. We call "Hankel index" the minimal rank of quadratic form (on extreme ray of the dual cone of SOS) which is not a point evaluation. In this talk, we introduce the Hankel index of variety and will discuss about a criterion to obtain the Hankel index of projected rational curves.