Effective Isometric Embeddings for Certain Hermitian Holomorphic Line Bundles

We consider bihomogeneous polynomials on complex Euclidean spacesthat are positive outside the origin and obtain effective estimateson certain modifications needed to turn them into squares ofnorms of vector-valued polynomials on complex Euclidean space.The corresponding results for hypersurfaces in complex Euclideanspaces are also proved. The results can be considered as Hermitiananalogues of Hilbert's seventeenth problem on representing apositive definite quadratic form on Rⁿ as a sum of squares ofrational functions. They can also be regarded as effective estimateson the power of a Hermitian line bundle required for isometricprojective embedding. Further applications are discussed.