On the Rank of Hermitian Polynomials and the SOS Conjecture

Abstract

Abstract 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 employ and extend the recent techniques developed for local orthogonal maps to study this conjecture and its generalizations to arbitrary signatures. We prove that for a given Hermitian form $\langle \cdot ,\cdot \rangle _\star $ on $\mathbb C^n$ of any signature with at least two nonzero eigenvalues, the ranks of the Hermitian polynomials of the form $A(z,\bar z)\|z\|^2_\star $ can only lie in certain disjoint intervals 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. These intervals can be explicitly calculated in terms of $n$ and the signature of $\langle \cdot ,\cdot \rangle _\star $. In addition, the number of such intervals and their widths are of the same order as those stated in the SOS conjecture.