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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.12951 |
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Table of Contents:
- A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $β$ is one of those roots, then within one unit of $β$ lies a root of the polynomial's derivative. If we define $r(β)$ to be the greatest possible distance between $β$ and the closest root of the derivative, then Sendov's conjecture claims that $r(β) \le 1$. In this paper, we conjecture that there is a constant $c>0$ so that $r(β) \le 1-cβ(1-β)$ for all $β\in [0,1]$. 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 $β$ is sufficiently close to $1$. In addition, we show that experimental data suggests that $c\approx0.233$.