The results in this chapter generally have the same flavor. We are given a polynomial whose values are positive on some set. We show that the polynomial agrees with a quotient of squared norms on that set, thus explaining the positivity. These results are striking applications of Theorem VII.1.1; this stabilization result shows that certain bihomogeneous polynomials are quotients of squared norms. See Remark VII.1.4 for an explanation of the term stabilization.

We state and discuss Theorem VII.1.1 in Section 1, but postpone its proof to Section 6. The proof uses the Hilbert space methods developed in this book; it combines properties of the Bergman kernel function for the unit ball in Cn with results about compact operators.

Stabilization for positive bihomogeneous polynomials

Recall that the term squared norm means for us a finite sum of squared absolute values of holomorphic polynomial functions. We have observed that a power of the squared Euclidean norm is itself a squared norm; ∥ z ∥2d = ∥ Hd (z) ∥2 = ∥ z⊗d ∥2. In this section we state and discuss Theorem VII.1.1. This result implies that a bihomogeneous polynomial, whose values away from the origin are positive, is necessarily a quotient of squared norms. We may always choose the denominator to be ∥ Hd (z) ∥2 for some d.

Let r be a real-valued bihomogeneous polynomial of degree 2m with matrix of coefficients (cαβ). Recall that Vm denotes the vector space of (holomorphic) homogeneous polynomials of degree m.