A stabilization theorem for Hermitian forms and applications to holomorphic mappings

We consider positivity conditions both for real-valued functions of several complex variables and for Hermitian forms. We prove a stabilization theorem relating these two notions, and give some applications to proper mappings between balls in different dimensions. The technique of proof relies on the simple expression for the Bergman kernel function for the unit ball and elementary facts about Hilbert spaces. Our main result generalizes to Hermitian forms a theorem proved by Polya [HLP] for homogeneous real polynomials, which was obtained in conjunction with Hilbert's seventeenth problem. See [H] and [R] for generalizations of Polya's theorem of a completely different kind. The flavor of our applications is also completely different.