A Scalar Optimization Proof of Sendov's Conjecture with Reduced Computer Assistance
Abstract
Let p be a complex polynomial of degree n≥2 whose zeros lie in the closed unit disk. Sendov's conjecture asserts that every zero of p lies within distance one of a zero of p'. Mazur's recent proof, and Tao's streamlined exposition of it, reduce a hypothetical counterexample to a scalar lower bound 1≤ F(η,α,n) together with two upper bounds for the endpoint parameter η and the uniform restriction 0<α≤ 17. We complete this scalar reduction by a two-stage optimization argument. First, F is nondecreasing in η, so η may be replaced by a piecewise polar envelope η∗(α). Writing s=(n−1)/2 converts the degree to a half-integer variable and gives F(η∗(α),α,n) = E(α,s) + κ(α,s) ∫01 t3 β̂(t;α,s) s−3/2 dt. On each half-unit α-slab, E and κ decrease with α, whereas β̂ increases. Convexity in t reduces all but the first mesh interval to eight explicit nodal terms. The first interval satisfies a uniform bound 3/200. Each nodal upper term is a strictly log-concave function of the half-integer s, so its global discrete maximum is certified by two adjacent ratio evaluations. A fixed finite certificate over the $34$ half-unit slabs gives F(η∗(α),α,n)<97/100<1, contradicting the scalar lower bound. The main contribution is a substantial reduction of the computer assistance required after the Mazur--Tao scalar reduction: all continuous optimization and the unbounded degree parameter are handled analytically, leaving only a small fixed collection of explicit one-variable inequalities.
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BibTeX
@misc{an2026scalar,
title = {{A Scalar Optimization Proof of Sendov's Conjecture with Reduced Computer Assistance}},
author = {Senjian An},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04246},
eprint = {2609.04246},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04246},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Scalar Optimization Proof of Sendov's Conjecture with Reduced Computer Assistance
AU - Senjian An
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.04246
UR - https://arxiv.org/abs/2609.04246
AB - Let p be a complex polynomial of degree n≥2 whose zeros lie in the closed unit disk. Sendov's conjecture asserts that every zero of p lies within distance one of a zero of p'. Mazur's recent proof, and Tao's streamlined exposition of it, reduce a hypothetical counterexample to a scalar lower bound 1≤ F(η,α,n) together with two upper bounds for the endpoint parameter η and the uniform restriction 0<α≤ 17. We complete this scalar reduction by a two-stage optimization argument. First, F is nondecreasing in η, so η may be replaced by a piecewise polar envelope η^∗(α). Writing s=(n−1)/2 converts the degree to a half-integer variable and gives F(η^∗(α),α,n) = E(α,s) + κ(α,s) ∫_0^1 t^3 β̂(t;α,s)^ s−3/2 dt. On each half-unit α-slab, E and κ decrease with α, whereas β̂ increases. Convexity in t reduces all but the first mesh interval to eight explicit nodal terms. The first interval satisfies a uniform bound 3/200. Each nodal upper term is a strictly log-concave function of the half-integer s, so its global discrete maximum is certified by two adjacent ratio evaluations. A fixed finite certificate over the $34$ half-unit slabs gives F(η^∗(α),α,n)<97/100<1, contradicting the scalar lower bound. The main contribution is a substantial reduction of the computer assistance required after the Mazur--Tao scalar reduction: all continuous optimization and the unbounded degree parameter are handled analytically, leaving only a small fixed collection of explicit one-variable inequalities.
ER -