An efficient sum of squares nonnegativity certificate for quaternary quartic

TL;DR

This paper studies nonnegativity certificates for 4-variable quartic polynomials, proposing an efficient sum of squares approach using quadratic forms and exploring conditions under which a single quadratic form suffices.

Contribution

It introduces an explicit sum of squares certificate for nonnegative quaternary quartics and analyzes the conditions for minimal quadratic form representations.

Findings

Explicit examples of non-sos quartics with non-sos discriminant

Demonstrates that multiplying by a quadratic form can yield sos representations

Conjectures that a single quadratic form may always suffice for sos decomposition

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 $\Delta$.