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

The paper studies Bernstein type inequalities for restrictions of holomorphic polynomials to graphs ${\mathrm{\Gamma }}_f\subset {\mathbb{C}}^{n+m}$ of holomorphic maps $f:{\mathbb{C}}^n\to {\mathbb{C}}^m$. 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.