Seeking a quadratic refinement of Sendov's conjecture

TL;DR

This paper proposes a refined version of Sendov's conjecture, introducing a constant-based bound on the distance between a root and the closest derivative root, supported by theoretical results and experimental data.

Contribution

It introduces a quadratic refinement of Sendov's conjecture and establishes bounds for specific polynomial classes, supported by theoretical proofs and experimental evidence.

Findings

Established bounds for degree 2 and 3 complex polynomials.

Derived bounds for degree 4 real polynomials and line-rooted polynomials.

Experimental data suggests the constant c is approximately 0.233.

Abstract

A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $\beta$ is one of those roots, then within one unit of $\beta$ lies a root of the polynomial's derivative. If we define $r(\beta)$ to be the greatest possible distance between $\beta$ and the closest root of the derivative, then Sendov's conjecture claims that $r(\beta) \le 1$. In this paper, we conjecture that there is a constant $c>0$ so that $r(\beta) \le 1-c\beta(1-\beta)$ for all $\beta \in [0,1]$. We find such constants for complex polynomials of degree $2$ and $3$, for real polynomials of degree $4$, for all polynomials whose roots lie on a line, for all polynomials with exactly one distinct critical point, and when $\beta$ is sufficiently close to $1$. In addition, we show that experimental data suggests that $c\approx0.233$.