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In this article, we combine sums of squares (SOS) and sums of nonnegativecircuit (SONC) forms, two independent nonnegativity certificates for realhomogeneous polynomials. We consider the convex cone SOS+SONC of forms thatdecompose into a sum of an SOS and a SONC form and study it from a geometricpoint of view. We show that the SOS+SONC cone is proper and neither closedunder multiplications nor under linear transformation of variables. Moreover,we present an alternative proof of an analog of Hilbert's 1888 Theorem for theSOS+SONC cone and prove that in the non-Hilbert cases it provides a propersuperset of both the SOS and the SONC cone. This follows by exploiting a newnecessary condition for membership in the SONC cone.

Geometrical Study of the Cone of Sums of Squares plus Sums of Nonnegative Circuits