Proof of Sendov's conjecture

TL;DR

This paper proves Sendov's conjecture, a long-standing mathematical hypothesis stating that for polynomials with zeros in the unit disk, each zero has a critical point within unit distance, confirming a key property of polynomial zeros.

Contribution

The paper provides a rigorous proof of Sendov's conjecture, resolving a problem that has remained open since the 1950s.

Findings

Sendov's conjecture is proven true for all polynomials with zeros in the unit disk.

The proof confirms the existence of a critical point within unit distance for each zero.

This result advances understanding of polynomial zero and critical point distributions.

Abstract

Sendov's conjecture, which was first introduced in the last 50s, asserts that if all the zeros of a polynomial $p$ lie in the closed unit disk then for each zero there must be a critical point of $p$ within unit distance. This paper confirms the conjecture.