On the rank of Hermitian polynomials and the SOS Conjecture

Let $z\in\mathbb C^n$ and $\|z\|$ be its Euclidean norm. Ebenfelt proposed a conjecture regarding the possible ranks of the Hermitian polynomials in $z,\bar z$ of the form $A(z,\bar z)\|z\|^2$, known as the SOS Conjecture, where SOS stands for "sums of squares". In this article, we employed and extended the recent techniques developed for local orthogonal maps to study this conjecture and its generalizations to arbitrary signatures. We proved that for a given Hermitian form $\langle\cdot,\cdot\rangle_\star$ on $\mathbb C^n$ of any signature with at least two non-zero eigenvalues, the ranks of the Hermitian polynomials of the form $A(z,\bar z)\|z\|^2_\star$ do not lie in a finite number of non-trivial gaps on the real line. Not only is this consistent with the SOS Conjecture, but it also demonstrates that in fact the conjecture qualitatively holds in all signatures except the trivial ones. In addition, these gaps can be explicitly calculated in terms of $n$ and the signature of $\langle\cdot,\cdot\rangle_\star$, and the number of such gaps and their sizes are of the same order as those stated in the SOS Conjecture.