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.

整个函数达到其最大模数的点集称为最大模数集。1951 年，海曼研究了这个集合在原点附近的结构。根据 Blumenthal 的工作，他表明，在接近零时，最大模量集由一组不相交的解析曲线组成，并为这些曲线的数量提供了上限。在本文中，除了泰勒级数系数满足某个简单代数条件的“小”集合外，我们为所有整个函数建立了这些曲线的确切数量。此外，我们给出了关于这个集合在原点附近的结构的新结果，并对最一般的情况做出了一个有趣的猜想。我们为次数小于 4 的多项式证明了这个猜想。