  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.
