Łojasiewicz exponent and pluricomplex Green functions on algebraic sets

The results obtained in this article form a necessary foundation for any prospective development of an extension of the pluripotential theory to algebraic sets. We focus on the key concept of pluricomplex Green function on such sets, as well as its interplay with different growth characterizations for holomorphic mappings. In particular, if f is a proper holomorphic mapping between two algebraic sets, then given a compact set K in the range of f , we show how to estimate the pluricomplex Green functions of K and of f −1(K) in terms of each other, the Łojasiewicz exponent of f and the growth exponent of f . This result leads to explicit examples of pluricomplex Green functions on algebraic sets. We also present an enhanced version of the Bernstein-Walsh polynomial inequality specific to algebraic sets. This article provides a theoretical framework for future investigations of the rate of polynomial approximation of holomorphic functions on algebraic sets in the style of Bernstein-Walsh-Siciak theorem.