Distribution of Zeros and Critical Points of a Polynomial, and Sendov’s Conjecture

Abstract

According to the Gauss–Lucas theorem, the critical points of a complex polynomial p(z):=sum_{j=0}^{n}a_{j}z^{j} where a_{j}inmathbb{C} always lie in the convex hull of its zeros. In this paper, we prove certain relations between the distribution of zeros of a polynomial and its critical points. Using these relations, we prove the well-known Sendov’s conjecture for certain special cases.