Complex variables analogues of Hilbert's seventeenth problem

We prove several theorems concerning the representation of Hermitian symmetric polynomials as quotients of squared norms of holomorphic polynomial mappings, thus providing complex variables analogues of Hilbert's seventeenth problem. We consider the space of Hermitian symmetric polynomials R on C ⁿ of degree at most d in z, with the Euclidean topology on the space of coefficients. We compare the collections of nonnegative polynomials P _d and quotients of squared norms Q_d. We prove, for d ≥ 2, that Q_d strictly contains the interior of P_d and is strictly contained in P_d. We provide a tractable precise description of Q_d in one dimension. We also give a necessary and sufficient condition in general in terms of F and G in the holomorphic decomposition R = ||F||² - ||G||². We provide many surprising examples and counterexamples and briefly discuss some applications.