Łojasiewicz exponent and pluricomplex Green function on algebraic sets

2022/12/20 by Bialas-Ciez, Leokadia, Klimek, Maciej
#32U15 #Complex Variables (math.CV) #FOS: Mathematics #Primary 32U35 #Secondary 32U05

paper · doi:10.48550/arxiv.2212.10119

Abstract

We study pluricomplex Green functions on algebraic sets. Let f be a proper holomorphic mapping between two algebraic sets. 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.

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