On the rank of Hermitian polynomials and the SOS Conjecture

Hilbert’s 17-th problem asked whether a non-negative polynomial in several real variables must be a sum of squares of rational functions. There is also a quantitative version of Hilbert’s 17th problem which asks how many squares are needed. D’Angelo extend this problem to more general case which is called Hermitian or complex variable analogues of Hilbert’s problem. 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 talk, we will introduce a dimension formula for local holomorphic mappings. As an application, we use this formula to study this conjecture and its generalizations to arbitrary signatures for a Hermition forms on $\mathbb C^n$. It is joint work with Sui-Chung Ng.