×

Positivity conditions for bihomogeneous polynomials. (English) Zbl 0886.32015

The authors continue the study of a complex variables version of Hilbert’s seventeenth problem by generalizing some of the results from D. W. Catlin and J. P. D’Angelo [Math. Res. Lett. 3, 149-166 (1996; Zbl 0858.32010)]. Given a bihomogeneous polynomial \(f\) of several complex variables that is positive away from the origin. There is an integer \(d\) so that \(||z||^{2d}f(z,\overline z)\) 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 squared Euclidean norm can be replaced by the 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 a compactness result is proved for an integral operator on such domains related to the Bergman kernel function.
The following results are proved: Suppose that \(\Omega\) is a bounded pseudoconvex domain in \(\mathbb{C}^n\) for which the \(\overline\partial-\)Neumann operator \(N\) is compact. Let \(M\) be a pseudodifferential operator of order 0. Then the commutator \([P,M]\) is compact on \(L^2(\Omega).\) Suppose that \(\Omega\) is a smoothly bounded pseudoconvex domain of finite type in \(\mathbb{C}^n\) with Bergman kernel \(B(z,\overline\zeta).\) Let \(g\) be a smooth function on \(\Omega\times\Omega\) that vanishes on the boundary diagonal. Then the operator on \(L^2(\Omega)\) with integral kernel \(gB\) is compact. Suppose that \(\Omega\) is a smoothly bounded pseudoconvex circled domain in \(\mathbb{C}^n\) of finite type. For each integer \(d,\) let \(\Phi^d=(\Phi^d_1,\dots,\Phi^d_N)\) denote an orthonormal basis for the homogeneous polynomials of degree \(d\) on \(\Omega.\) Let \(f\) be a bihomogeneous polynomial that is positive away from the origin. Then there is an integer \(d_0\) (depending on \(f\)) such that, for each \(d\geq d_0,\) there is homogeneous polynomial mapping \(h\) such that \(||\Phi^d(z)||^2f(z,\overline z)=||h(z)||^2.\) The obtained results are interpreted in terms of Hermitian line bundles over complex projective space.

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators

Citations:

Zbl 0858.32010
PDFBibTeX XMLCite
Full Text: DOI arXiv