On the Sendov conjecture for polynomials with simple zeros

TL;DR

This paper provides a partial proof for the Sendov conjecture specifically for polynomials with simple zeros, advancing understanding of the zero distribution in relation to the conjecture.

Contribution

It offers a new partial result confirming the Sendov conjecture for polynomials with simple zeros, a case not fully resolved before.

Findings

Partial validation of Sendov conjecture for simple zeros

Identification of conditions under which the conjecture holds

Progress towards a complete proof for all polynomials

Abstract

The Sendov conjecture asserts that if all the zeros of a polynomial p lie in the closed unit disk then there must be a zero of p ' within unit distance of each zero. In this paper we give a partial result when p has simple zeros.