An efficient sum of squares certificate for 4-ary 4-ic

We show that for any non-negative quaternary quartic form $f$ there exists a product of two non-negative quadrics $q$ so that $qf$ is a sum of squares (s.o.s.) of quartics. This is a much better upper bound on the degree of a multiplier needed to demonstrate nonnegativity of $f$ via an s.o.s. decomposition than known previously; similar almost tight bounds are only known for ternary forms.

As a step towards deciding whether it is sufficient to use a quadratic multiplier $q$, we show that there exist non-s.o.s. non negative ternary sextics $ac-b^2$, with $a$, $b$, $c$ of degrees 2, 3, 4, respectively.