具有Baker省略值的指数多项式
Exponential polynomials with Baker omitted value
- Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram（印度信息技术设计与制造学院（坎切普拉姆））
- Indian Institute of Technology Bhubaneswar（印度理工学院布巴内斯瓦尔分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究一类指数多项式在无穷远点邻域原像的拓扑，证明其连通且为无穷连通域，无穷远点为Baker省略值，并推广了Das和Nayak的方法，同时证明无Baker游荡域，所有Fatou分量单连通。
AI中文摘要：
我们考虑形如$F_l(z)=P_1(z)\exp(Q_1(z))+P_2(z)\exp(Q_2(z))+\cdots+P_{l-1}(z)\exp(Q_{l-1}(z))+P_l(z)$的指数多项式，其中$l\geq2$，每个$Q_i$是非常数多项式，每个$P_i$（$i=1,2,\ldots,l-1$）是多项式（可能为常数），而$P_l$是非常数多项式。本文研究$F_l$在无穷远处本质奇点邻域的原像的拓扑性质。我们证明，对于无穷远的每个邻域$D$，集合$F_l^{-1}(D)$是连通的，实际上是一个无穷连通域，且其所有边界分量都是有界的。在这种情况下，无穷远点被称为该函数的Baker省略值。我们的方法显著推广了Das和Nayak（见Complex Var. Elliptic Equ. 70:1831-1847, 2025）所发展的方法，为研究这一现象提供了更广泛的框架。我们进一步证明，这些函数不存在任何Baker游荡域，因此每个Fatou分量都是单连通的。最后，我们提出了一些由此工作产生的问题作为结论。
英文摘要：
We consider exponential polynomials of the form $F_l(z)=P_1(z)\exp(Q_1(z))+P_2(z)\exp(Q_2(z))+\cdots+P_{l-1}(z)\exp(Q_{l-1}(z))+P_l(z)$, where $l\geq2$, each $Q_i$ is a non-constant polynomial and each $P_i$, $i=1,2,3\ldots,l-1$, is a polynomial (possibly constant), while $P_l$ is a non-constant polynomial. This article studies the topology of the preimages under $F_l$ of the neighborhoods of the essential singularity at infinity. We prove that for each disk $D$ (with respect to the spherical metric) centered at infinity, the set $F_l ^{-1} (D) $ is connected and, in fact, is an infinitely connected domain with all its boundary components bounded. In such a situation, the point at $\infty$ is called the Baker omitted value of the function. Our methods extend those developed by Das and Nayak for the function $\exp(z)+P(z)$ for any non-constant polynomial $P$ (Complex Var. Elliptic Equ. 70 (2025), 1831-1847), providing a broader framework for studying this phenomenon. We further show that these functions do not admit any Baker wandering domain. Finally, we conclude by posing some problems arising out of this work.
