Extremal Problems for Polynomials in the Complex Plane

Abstract

This is a survey on some particular polynomial problems that are related to complex analogs of Rolle’s theorem or to the Bernstein majorization theorem that implies the well-known estimate for the derivative of a complex polynomial on the disk. The main topic, however, is Sendov’s conjecture about the critical points of algebraic polynomials. Despite the numerous attempts to verify the conjecture, it is not settled yet and remains as one of the most challenging problems in the analytic theory of polynomials. We also discuss the mean value conjecture of Smale and point out to certain relation between these two famous open problems. Finally, we formulate a conjecture that seems to be a natural complex analog of Rolle’s theorem and contains as a particular case Smale’s conjecture.