A Proof of Sendov’s Conjecture

The zeros of the derivative of a complex polynomial p are functions of the zeros of p itself. In general we do not know explicit expressions for these functions. So approximate localizations of the derivative zeros in terms of the given zeros of p are of interest. A question of this type is the famous conjecture of Bl. Sendov which goes back to 1959 and took place in Hayman’s booklet on problems in Complex Analysis (1967, [2], by a misunderstanding, there it was named after Ilieff). It has been one of the most important problems in the zone between Function Theory and Algebraic Geometry in the least 40 years. This conjecture states: Let p ∈ C [z] be a polynomial of degree n > 1 having all zeros z1, . . . , zn in the closed unit disk E. Does there exist for every zj some ζ with |zj − ζ | ≤ 1 and p (ζ) = 0 ? For a history of the conjecture and a list of the numerous (about 100) publications on it, most of them in famous international journals, see the recent article of Bl. Sendov [9] or the survey of Schmeisser [8]. In this paper we give a proof of this question. By Pn we denote the class of all monic polynomials of degree n > 1 having all its zeros in E. For the following we fix some polynomial p ∈ Pn with the factorization