I will consider a very natural class of functions,
called Hermitian algebraic, that generalize Hermitian polynomials. I will present a Hermitian analog, introduced by J. D'Angelo but already considered implicitly by D. Quillen, of Hilbert's 17th problem:
"When is a non-negative Hermitian algebraic function a quotient of squared norms of holomorphic mappings?" I will state a complete solution to the
problem, and then prove the result in a special (previously known) case. The proof I will present is new and simpler than the original proof. If time permits, I will indicate how one treats the general case.