An efficient sum of squares nonnegativity certificate for quaternary quartic

Abstract

For any 4-variate quartic form $f\geq 0$ (i.e. $f$ nonnegative, homogeneous polynomial of degree $4$ with real coefficients) there exist quadratic forms $q$ and $q'$ so that $qq'f$ is a sum of squares (s.o.s.) of quartics, by reducing to the case of $f=au^2+2bu+c$ with $a$, $b$, $c$ $3$-variate forms of degrees 2, 3, 4, respectively, and invoking on its discriminant $\Delta=ac-b^2$ a theorem by Hilbert (1893) asserting that for any ternary sextic $h\geq 0$ there exists a quadric $q''$ so that $q''h$ is s.o.s. of quartics. Towards deciding whether just one $q$ always suffices to make $qf$ a s.o.s, we give explicit examples of non-s.o.s. $f=au^2+2bu+c\geq 0$ with non-s.o.s. $\Delta$. However, in all these examples $af$ are s.o.s. That is, the straightforward s.o.s. decomposition via Hilbert (1893) need not be the best possible. While it remains open whether one $q$ always suffices (and we conjecture that $q=a$ suffices), we describe how the existence of such $q$ is related to particular types of s.o.s. decompositions for $\Delta$.