Georgia Institute of Technology College of Sciences

A globally nonnegative polynomial F is called stubborn if no odd power of F is a sum of squares. We develop a new invariant of a singularity of a form (homogeneous polynomial) in 3 variables, which allows us to conclude that if the sum of these invariants over all zeroes of a nonnegative form is large enough, then the form is stubborn. As a consequence, we prove that if an extreme ray of the cone of nonnegative ternary sextics is not a sum of squares, then all of its odd powers are also not sums of squares, and we provide more examples of this phenomenon for ternary forms in higher degree. This is joint work with Khazhgali Kozhasov and Bruce Reznick.