AI 中文总结
本文基于稳定残差原理，给出Erdős问题973否定答案的另一种证明，得到rₙ的下界及相关极限性质，通过投影牛顿恒等式转化结果，完成残差估计。
AI 中文摘要
我们给出Erdős问题973否定答案的另一种证明。具体而言，对任意λ>0，存在整数N=N(λ)，使得对所有整数n≥N及所有满足|z_j|≥1的复数z₁,…,zₙ，即使不满足z₁=1的归一化条件，也有max₂≤k≤n+1|∑ⱼ₌₁ⁿ zⱼᵏ|>exp(-λn。该结论此前已由Luo、Yang和Zhu得到，本文的贡献是基于稳定残差原理的另一种证明。设rₙ为所有次数为n、满足P(0)=1且所有零点在闭单位圆盘内的多项式P，对应的‖P′−P′(0)P‖_{H²}/‖P‖_{H²}的下确界。我们证明存在绝对常数D>0和N₀，使得对所有n≥N₀，有rₙ≥exp(-D n^(2/3)(log(en))^(2/3))，故rₙ^(1/n)→1。一个精确的投影牛顿恒等式将该结果转化为外幂和问题。残差估计通过在边界最大值处归一化、积分一阶常微分方程分离零点，并利用有限多项式检验排除所得矩分布得到。
英文摘要
Let $r_n$ be the infimum of \[ \frac{\lVert P'-P'(0)P\rVert_{H^2}}{\lVert P\rVert_{H^2}} \] over all degree-$n$ polynomials $P$ satisfying $P(0)=1$ whose zeros lie in the closed unit disk. We prove the quantitative residual bound \[ r_n\geq \exp\!\bigl(-(1+o(1))\sqrt n\log n\bigr) \qquad(n\to\infty). \] As an application, we answer Erdős Problem 973 on exterior power sums in the negative, in a form quantitatively stronger than the answer first obtained by Luo, Yang, and Zhu.
Comments10 pages. Xiaojun Tan and Qihang Wang contributed equally. Substantially revised: the paper now centers on residual bounds for Schur-stable polynomials and proves r_n >= exp(-(1+o(1)) sqrt(n) log n). Erdos Problem 973 is presented as an application
