{\L}ojasiewicz exponent and pluricomplex Green function on algebraic sets

TL;DR

This paper investigates pluricomplex Green functions on algebraic sets, establishing estimates related to proper holomorphic mappings, and introduces an improved Bernstein-Walsh inequality for algebraic sets, advancing polynomial approximation theory.

Contribution

It provides new estimates for pluricomplex Green functions under holomorphic mappings and presents an enhanced Bernstein-Walsh inequality specific to algebraic sets.

Findings

Estimates of Green functions in terms of the Łojasiewicz exponent and growth exponent.

Explicit examples of pluricomplex Green functions on algebraic sets.

An improved Bernstein-Walsh polynomial inequality for algebraic sets.

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 {\L}ojasiewicz 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.