A stabilization theorem for Hermitian forms and applications to holomorphic mappings

Abstract

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.

Polynomial Optimization on Odd-Dimensional Spheres

“…This result was discovered independently by the first author and Catlin [6] in conjunction with the first author's work on proper mappings between balls in different dimensions. The proof in [6] uses the Bergman projection and some facts about compact operators, and it generalizes to provide an isometric imbedding theorem for certain holomorphic vector bundles [7].…”

“…The proof in [6] uses the Bergman projection and some facts about compact operators, and it generalizes to provide an isometric imbedding theorem for certain holomorphic vector bundles [7]. It is worth noting that the integer k and the number of components of h can be arbitrarily large, even for polynomials p of total degree four in two variables.…”

“…Using a process of bihomogenization, Catlin and the first author (see [6], [8] and [9]) proved that if p is arbitrary (not necessarily bihomogeneous) and strictly positive on the sphere, then p agrees with a squared norm on the sphere; in other words, p ∈ Σ …”

Hermitian Symmetric Polynomials and CR Complexity

“…In (10) the factors of 1 + t 12 have signature pairs (2, 1) and (6,3), and hence each defines an indefinite form. Their product determines a Hermitian form with signature pair (2, 0).…”

“…The general answer seems to be complicated, and hence we are satisfied with the stability result. Theorem 7.1 determines what happens for each pair (A, B), with the only exceptions being (3,2) and (2,3). Analyzing the cases of small rank is difficult.…”