【摘要】
Hilbert’s 17th 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 a more general case which is called Hermitian or complex variable analogues of Hilbert's problem. Ebenfelt proposed a conjecture regarding the possible ranks of the Hermitian polynomials, known as the SOS Conjecture, where SOS stands for "sums of squares". This conjecture implies GAP conjecture. In this talk, we study this conjecture and its generalizations to arbitrary signatures for Hermitian forms on $C^n$ . It is a joint work with Sui-Chung Ng.
