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Introduction.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.We begin by describing our main result. Let C n denote complex Euclidean space of n dimensions, with complex Euclidean squared norm ||z|| 2 = n j=1 |z j | 2 . Suppose that f : C n → R is a polynomial in the variables z and z, and that it is homogeneous of the same degree m in each of these variables. We write f (z, z) = |α|=m |β|=m c αβ z α z β . The condition that f take only real values is equivalent to c αβ = c βα . We call the Hermitian matrix (c αβ ) the underlying matrix of coefficients. Then we have the following conclusion. The function f achieves a positive minimum value on the unit sphere if and only if there is an integer d so that the matrix (E µν ) is positive definite, where (E µν ) is the underlying matrix of coefficients for the function ||z|| 2d f (z, z), that isConsequently there is a homogeneous holomorphic vector-valued polynomial g such thatWhen (c αβ ) is itself positive definite, then f must be positive on the sphere; this is the case d = 0. In general the smallest possible value for the integer d depends on the original underlying matrix of coefficients. We note that, once the form is positive for some d, it remains positive for all larger values, and this suggests the name "stabilization". In case the original underlying matrix of coefficients is diagonal, this theorem implies the classical theorem of Polya in the real case. Even in the diagonal case the smallest possible integer d can be arbitrarily large. See Example 2 and Remark 1.

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IntroductionIn this paper we continue our study of a complex variables version of Hilbert's seventeenth problem by generalizing some of the results from [CD]. Given a bihomogeneous polynomial f of several complex variables that is positive away from the origin, we proved that there is an integer d so that ||z|| 2d f (z, z) is the squared norm of a holomorphic mapping. Thus, although f may not itself be a squared norm, it must be the quotient of squared norms of holomorphic homogeneous polynomial mappings. The proof required some operator theory on the unit ball. In the present paper we prove that we can replace the squared Euclidean norm by squared norms arising from an orthonormal basis for the space of homogeneous polynomials on any bounded circled pseudoconvex domain of finite type. To do so we prove a compactness result for an integral operator on such domains related to the Bergman kernel function. Recall that the Bergman kernel function B for a domain Ω is the integral kernel for the operator P that projects L 2 (Ω) to the closed subspace A 2 (Ω) of holomorphic functions in L 2 (Ω). We prove the following results. Proposition 1. Suppose that Ω is a bounded pseudoconvex domain in C n for which the ∂-Neumann operator N is compact. Let M be a pseudodifferential operator of order 0. Then the commutator [P, M ] is compact on L 2 (Ω). Theorem 1. Suppose that Ω is a smoothly bounded pseudoconvex domain of finite type in C n , with Bergman kernel B(z, ζ). Let g be a smooth function on Ω × Ω that vanishes on the boundary diagonal. Then the operator on L 2 (Ω) with integral kernel gB is compact.Theorem 2. Suppose that Ω is a smoothly bounded pseudoconvex circled domain in C n of finite type. For each integer d, let Φ d = (Φ