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

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.