Searching for a sharp version of the Iliev-Sendov conjecture

Abstract

The well-known Iliev-Sendov conjecture states that if p(z) is a polynomial of degree n ≥ 2, with all zeroes in the (closed) unit disc, there is at least one zero of the derivative p(z) within unit length from each given zero of the polynomial. The purpose of this paper is to initiate a search for a sharp version of the conjecture and to inspire further work towards identifying necessary subsets of the minimal regions containing at least one zero of the derivative. The approach is to define such a region En(a) for each point a in the unit disc and for each degree n ≥ 2. We establish that E2(a) = {z : |z − a 2| ≤ 12} and En(0) = {z : |z| ≤ 1n−1√n}. Furthermore, we determine that {z : |z − a 2| ≤√12−3|a|26 } ⊆ E3(a) if |a| ≤ 1, and that {z = (x, y) : (x − a 2 )2 + 11−a2 y2 ≤ 14} ⊆ En(a) if 0 ≤ a < 1. However, the main purpose of this paper is to introduce and motivate the definition of the minimal regions En(a), and to inspire further work towards identifying these regions.