EFFECTIVE ISOMETRIC EMBEDDINGS FOR CERTAIN HERMITIAN HOLOMORPHIC LINE BUNDLES

Abstract

We consider bihomogeneous polynomials on complex Euclidean spaces that are positive outside the origin and obtain effective estimates on certain modifications needed to turn them into squares of norms of vector-valued polynomials on complex Euclidean space. The corresponding results for hypersurfaces in complex Euclidean spaces are also proved. The results can be considered as Hermitian analogues of Hilbert's seventeenth problem on representing a positive definite quadratic form on $\mathbb{R}^n$ as a sum of squares of rational functions. They can also be regarded as effective estimates on the power of a Hermitian line bundle required for isometric projective embedding. Further applications are discussed.