论文标题
Hayman关于最大模量集的结果
On a result of Hayman concerning the maximum modulus set
论文作者
论文摘要
整个函数实现其最大模量的一组点称为最大模量集。 1951年，海曼研究了该集合附近的结构。在Blumenthal的工作之后，他表明，接近零，最大模量集由一系列不连接分析曲线组成，并为这些曲线的数量提供了上限。在本文中，我们为所有函数建立了这些曲线的确切数量，除了一个“小”集，其Taylor系列系数满足某些简单的代数条件。 此外，我们给出了有关该集合在原点附近的结构的新结果，并就最一般的情况做出了有趣的猜想。我们证明了该程度小于四的多项式的猜想。
The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a "small" set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concerning the structure of this set near the origin, and make an interesting conjecture regarding the most general case. We prove this conjecture for polynomials of degree less than four.
