Positivity conditions for bihomogeneous polynomials

Abstract

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

Compactness of the solution operator to ∂¯ in weighted L2-spaces

“…For example, compactness of N q implies global regularity in the sense of preservation of Sobolev spaces [32]. Also, the Fredholm theory of Toeplitz operators is an immediate consequence of compactness in the ∂-Neumann problem [6,27,50]. There are additional ramifications for certain C * -algebras naturally associated to a domain in C n [41].…”

“…In [6] it is shown that compactness of the ∂-Neumann operator implies compactness of the commutator [P , M], where P is the Bergman projection and M is pseudodifferential operator of order 0. In [17] it is shown that compactness of the canonical solution operator to ∂ restricted to (0, 1)-forms with holomorphic coefficients implies compactness of the commutator [P , M] defined on the whole L 2 (Ω).…”

Some geometric properties of successive difference substitutions

“…When deciding nonnegativity of polynomials using traditional methods, complexities of algorithms increase rapidly as variable numbers and degrees of the polynomials increase [4][5][6][7][8][9].…”

“…, c r n2 (except all zeros) have same signs in the σ coordinate system. Analogously, by finite steps of SDS, (8) …”

The canonical solution operator of $ {\bar \partial } $ on weighted spaces with holomorphic coefficients

“…Consequently with these formulas boundedness and compactness of N and S are equivalent. For compactness see [1].…”