Exponential polynomials with Baker omitted value

Abstract

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 $l\ge 2$, each $Q_i$ is a non-constant polynomial and each $P_i$, $i=1,2,3\dots ,l-1$, 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.

arXiv comment25 pages, 3 figures, Comments are welcome