Exponential polynomials with Baker omitted value
Abstract
We consider exponential polynomials of the form Fl(z)=P1(z)exp(Q1(z))+P2(z)exp(Q2(z))+⋯+Pl−1(z)exp(Ql−1(z))+Pl(z), where l≥2, each Qi is a non-constant polynomial and each Pi, i=1,2,3…,l−1, is a polynomial (possibly constant), while Pl is a non-constant polynomial. This article studies the topology of the preimages under Fl 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 ∞ 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.
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BibTeX
@misc{das2026exponential,
title = {{Exponential polynomials with Baker omitted value}},
author = {Sukanta Das and Subhasis Ghora and Tarakanta Nayak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24154},
eprint = {2609.24154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24154},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exponential polynomials with Baker omitted value
AU - Sukanta Das
AU - Subhasis Ghora
AU - Tarakanta Nayak
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.24154
UR - https://arxiv.org/abs/2609.24154
AB - We consider exponential polynomials of the form F_l(z)=P_1(z)exp(Q_1(z))+P_2(z)exp(Q_2(z))+⋯+P_l−1(z)exp(Q_l−1(z))+P_l(z), where l≥2, each Q_i is a non-constant polynomial and each P_i, i=1,2,3…,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 ∞ 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.
ER -