关于平面上渐近极值多项式列的零点
On the zeros of asymptotically extremal polynomial sequences in the plane
摘要：
设$E$是复平面上正对数容量的紧集，$\{P_n(z)\}_{1}^{\infty}$是$E$的渐近极值首一多项式序列，在此意义下，\begin{equation*}%\label{} \limsup_{n\to\infty}\|P_n\|_E^{1/n}\le\mathrm{cap}(E). \end{equation*}.本文的目的是给出$E$上$\{P_n\}$的零点的归一化计数测度的（全）序列在弱收敛的充分几何条件.利用Gardiner和Pommerenke关于测度平衡的一个论证，我们证明了这是正确的，例如，如果$E$的多项式凸包的内部具有单个分量并且该分量的边界具有“内角”，（更一般地，“非凸奇点”）。这一简单的事实迄今尚未在文献中得到充分强调。作为应用，改进了某些特殊多项式序列零点分布的已知结果。
英文摘要：
Let $E$ be a compact set of positive logarithmic capacity in the complex plane and let $\{P_n(z)\}_{1}^{\infty}$ be a sequence of asymptotically extremal monic polynomials for $E$ in the sense that \begin{equation*}%\label{} \limsup_{n\to\infty}\|P_n\|_E^{1/n}\le\mathrm{cap}(E). \end{equation*} The purpose of this note is to provide sufficient geometric conditions on $E$ under which the (full) sequence of normalized counting measures of the zeros of $\{P_n\}$ converges in the weak-star topology to the equilibrium measure on $E$, as $n\to\infty.$ Utilizing an argument of Gardiner and Pommerenke dealing with the balayage of measures, we show that this is true, for example, if the interior of the polynomial convex hull of $E$ has a single component and the boundary of this component has an "inward corner" (more generally, a "non-convex singularity"). This simple fact has thus far not been sufficiently emphasized in the literature. As applications we mention improvements of some known results on the distribution of zeros of some special polynomial sequences.
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