斯梅尔中值猜想及其关于复多项式的对偶猜想
Smale's Mean Value Conjecture and its Dual Conjecture for Complex Polynomials
浏览论文内容
中文总结 AI 辅助
本文针对复多项式的斯梅尔中值猜想及其对偶猜想展开研究，通过改进Koebe型定理与马尔可夫型不等式，将对偶猜想的最优无条件下界从1/d²提升至(d−1/2)^(1/d)/d²。
中文摘要 AI 辅助
为证明斯梅尔中值猜想，可将研究范围限定于一类明确描述的所谓单叶标准化多项式。对于首项系数衰减足够缓慢的单叶标准化多项式，我们证明当次数d趋于无穷时，猜想的上界1渐近成立。关键且新颖的输入是Cunningham的改进型Koebe 1/4型定理，该定理针对像具有有界对数容量的单叶函数量身定制。对于对偶中值猜想，我们在多项式莱姆尼斯科特界定的多项式单叶区域上改进了Eremenko的马尔可夫型不等式。由此，我们将Dubinin给出的所有d≥2时的最优无条件下界1/d²，改进为(d−1/2)^(1/d)/d²。强化后的马尔可夫型不等式是本文的主要技术贡献之一，其证明结合了多项式莱姆尼斯科特对数容量相关极值问题的解，以及Eremenko和Hayman引入的拟共形变形方法，以确立Erdos、Herzog和Piranian关于极大莱姆尼斯科特长度问题中产生的极值多项式莱姆尼斯科特的连通性。
英文摘要
For the purposes of proving Smale's mean value conjecture, one may restrict consideration to an explicitly described family of so-called Schlicht normalized polynomials. For Schlicht normalized polynomials whose leading coefficient decays sufficiently slowly, we show that the conjectured upper bound $1$ becomes asymptotically valid as the degree $d$ tends to infinity. The key and novel input is a refined Koebe $\frac{1}{4}$-type theorem of Cunningham, tailored to Schlicht functions whose images have bounded logarithmic capacity. For the dual mean value conjecture, we obtain an improvement of Eremenko's Markov-type inequality for regions bounded by polynomial lemniscates on which the polynomial is univalent. As a consequence, we improve the best known unconditional lower bound $\frac{1}{d^2}$ of Dubinin to $\frac{(d-\frac{1}{2})^{1/d}}{d^2}$ for all $d \ge 2$. The strengthened Markov-type inequality constitutes one of the main technical contributions of this paper. Its proof combines the solution to a related extremal problem for the logarithmic capacity of polynomial lemniscates, with the quasi-conformal deformation method introduced by Eremenko and Hayman to establish the connectedness of extremal polynomial lemniscates arising in the Erdos, Herzog and Piranian's problem on maximal lemniscate length.
