Extremal growth of polynomials (Q2204112)

This expository article addresses the determination of a sharp upper bound on the value \(\vert p(z)\vert \) when \(p\) is a holomorphic polynomial whose values on a specified compact set \(K\) have absolute value no larger than \(1\), and the point \(z\) lies outside \(K\). A main point of the exposition is to avoid using refined methods from pluripotential theory. \par The authors solve the problem using elementary techniques when \(K\) is a ball with respect to an arbitrary norm in either real or complex Euclidean space, a real simplex in real Euclidean space, and even a general convex body in real Euclidean space. They also determine the Siciak-Zaharjuta extremal function for a real simplex in complex Euclidean space and for a real Euclidean ball in complex Euclidean space.