Effective Ljasiewicz inequality for arithmetically defined varieties and a geometric application to bihomogeneous polynomials

Abstract

We establish two versions of effective Lojasiewicz inequality for arithmetically defined affine varieties. As an application, we consider bihomogeneous polynomials on the complex Euclidean space which are positive along the affine cone of an arithmetically defined projective variety, and we obtain effective estimates on certain modifications needed to turn them into sums of squares of pointwise norms of homogeneous polynomials. The latter can be interpreted as an effective result on isometric embeddings for the associated indefinite Hermitian holomorphic line bundles. © 2009 World Scientific Publishing Company.