Bernstein Type Inequalities for Restrictions of Polynomials to Complex Submanifolds of ${\mathbf {\mathbb C^N}}$

TL;DR

This paper investigates Bernstein inequalities for holomorphic polynomial restrictions to complex submanifolds, analyzing exponents and identifying classes of graphs with optimal inequality properties.

Contribution

It introduces general properties of Bernstein inequality exponents and characterizes classes of graphs with optimal or polynomial growth exponents.

Findings

Established properties of Bernstein inequality exponents

Identified classes of graphs with optimal exponents

Described polynomial growth behavior of exponents

Abstract

The paper studies Bernstein type inequalities for restrictions of holomorphic polynomials to graphs $\Gamma_f\subset\mathbb C^{n+m}$ of holomorphic maps $f:\mathbb C^n\rightarrow\mathbb C^m$. We establish general properties of exponents in such inequalities and describe some classes of graphs admitting Bernstein type inequalities of optimal exponents and of exponents of polynomial growth.