An efficient sum of squares nonnegativity certificate for quaternary quartic

Abstract: We show that for any non-negative 4-ary quartic form $f$ there exists a product of two non-negative quadrics $q$ and $q'$ so that $qq'f$ is a sum of squares (s.o.s.) of quartics. As a step towards deciding whether just one $q$ always suffices to make $qf$ a s.o.s, we show that there exist non-s.o.s. non-negative 3-ary sextics $ac-b^2$, with $a$, $b$, $c$ of degrees 2, 3, 4, respectively.