Expansivity theory and Sendov's conjecture

TL;DR

This paper introduces expansivity of polynomial tuples, reformulates Sendov's conjecture within this framework, proves weak variants, and applies the theory to differential equations, ultimately providing a proof of Sendov's conjecture.

Contribution

It develops the concept of expansivity for polynomial tuples, reformulates Sendov's conjecture in this context, and proves some weak variants, leading to a proof of the conjecture.

Findings

Weak variants of Sendov's conjecture are proven.

A new framework for analyzing polynomial zeros and critical points is established.

The theory is applied to differential equations to demonstrate its utility.

Abstract

In this paper we introduce and develop the concept of expansivity of a tuple whose entries are elements from the polynomial ring $\mathbb{C}[x]$. As an inverse problem, we examine how to recover a tuple from the expanded tuple at any given phase of expansion. We convert the celebrated Sendov conjecture concerning the distribution of zeros of polynomials and their critical points into this language and prove some weak variants of this conjecture. We also apply this to the existence of solutions to differential equations. In particular, we show that a certain system of differential equation has no non-trivial solution. As an application we give a proof of Sendov's conjecture. We start by establishing the uniformly diminishing state of the mass of an expansion.