AI 中文总结
本研究完成森多夫猜想的标量优化证明，通过解析处理连续优化与无界次数参数，仅保留少量显式单变量不等式，大幅降低了所需的计算机辅助。
AI 中文摘要
设p为次数n≥2的复多项式，其零点位于闭单位圆盘内。森多夫猜想断言，p的每个零点都与p'（导数多项式）的某个零点的距离不超过1。Mazur近期的证明以及陶哲轩对该证明的精简阐述，将假设的反例归约为一个标量下界1≤F(η,α,n)，以及端点参数η和均匀约束0<α≤17的两个上界。我们通过两阶段优化论证完成该标量归约：首先，F关于η非递减，因此η可替换为分段极包络η*(α)；令s=(n-1)/2可将次数转换为半整数变量，得到F(η*(α),α,n)=E(α,s)+κ(α,s)∫₀¹t³β̂(t;α,s)^(s-3/2)dt。在每个半单位α-板上，E和κ随α减小，而β̂随α增大；t的凸性将除第一个网格区间外的所有区间简化为8个显式节点项，第一个区间满足均匀界3/200；每个节点上项是半整数s的严格对数凹函数，其全局离散最大值可通过两个相邻比值评估验证。对34个半单位板的固定有限验证给出F(η*(α),α,n)<97/100<1，与标量下界矛盾。本研究的主要贡献是大幅减少Mazur-Tao标量归约后所需的计算机辅助：所有连续优化和无界次数参数均通过解析方式处理，仅留下少量固定的显式单变量不等式。
英文摘要
Let $p$ be a complex polynomial of degree $n\ge2$ 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 \[ 1\le F(η,α,n) \] together with two upper bounds for the endpoint parameter $η$ and the uniform restriction $0<α\le 17$. 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 $η^{\ast}(α)$. 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 $α$-slab, $E$ and $κ$ decrease with $α$, whereas $\widehatβ$ 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.
