Roots of polynomials and the Sendov's conjecture

TL;DR

This paper proves that all roots of polynomials, including simple roots, satisfy Sendov's conjecture, thereby confirming the conjecture's validity for all roots.

Contribution

The paper provides a proof that simple roots satisfy Sendov's conjecture, extending the trivial case of multiple roots to all roots of polynomials.

Findings

Simple roots satisfy Sendov's conjecture

Multiple roots trivially satisfy Sendov's conjecture

Sendov's conjecture holds for all roots of polynomials

Abstract

In this paper we first prove that a simple root of a polynomial satisfies the Sendov's conjecture. As the multiple roots trivially satisfy the Sendov's conjecture we conclude that the Sendov's conjecture holds true.