Polynomials with zeros and small norm on curves

This paper considers the problem of how zeros lying on the boundary of a domain influence the norm of polynomials (under the normalization that their value is fixed at a point). It is shown that $k$ zeros raise the norm by a factor $(1+ck/n)$ (where $n$ is the degree of the polynomial), while $k$ excessive zeros on an arc compared to $n$ times the equilibrium measure raise the norm by a factor $\exp (ck^2/n)$. These bounds are sharp, and they generalize earlier results for the unit circle which are connected to some constructions in number theory. Some related theorems of Andrievskii and Blatt will also be strengthened.