Macaulay representation of the prolongation matrix and the SOS conjecture

Let $z\in {\mathbb{C}}^n$, and let $A(z,\bar{z})$ be a real valued diagonal bihomogeneous Hermitian polynomial such that $A(z,\bar{z})\mathrm{\parallel }z{\mathrm{\parallel }}^2$ is a sum of squares, where $\mathrm{\parallel }z\mathrm{\parallel }$ denotes the Euclidean norm of $z$. In this paper, we provide an estimate for the rank of the sum of squares $A(z,\bar{z})\mathrm{\parallel }z{\mathrm{\parallel }}^2$ when $A(z,\bar{z})$ is not semipositive definite. As a consequence, we confirm the SOS conjecture proposed by Ebenfelt for $4\le n\le 6$ when $A(z,\bar{z})$ is a real valued diagonal (not necessarily bihomogeneous) Hermitian polynomial, and we also give partial answers to the SOS conjecture for $n\ge 7$.