A Scalar Optimization Proof of Sendov's Conjecture with Reduced Computer Assistance

Abstract

Let $p$ be a complex polynomial of degree $n\ge 2$ 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 $p^{\prime }$. Mazur's recent proof, and Tao's streamlined exposition of it, reduce a hypothetical counterexample to a scalar lower bound \[ 1\le F(η,α,n) \] together with two upper bounds for the endpoint parameter $\eta$ and the uniform restriction $0<\alpha \le 17$. We complete this scalar reduction by a two-stage optimization argument. First, $F$ is nondecreasing in $\eta$, so $\eta$ may be replaced by a piecewise polar envelope ${\eta }^{\ast }(\alpha )$. Writing $s=(n-1)/2$ converts the degree to a half-integer variable and gives \[ F(η^{\ast}(α),α,n) = E(α,s) + κ(α,s) \int_0^1 t^3 \widehatβ(t;α,s)^{\,s-\frac{3}{2}} \,dt. \] On each half-unit $\alpha$-slab, $E$ and $\kappa$ decrease with $\alpha$, whereas $\hat{\beta }$ 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 \[ F(η^{\ast}(α),α,n)<\frac{97}{100}<1, \] 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.