Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

An exponential polynomial is a finite linear sum of terms P(z)e(Q(z)), where P(z) and Q(z) are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg Polya's paper published in 1920, while the latest results have come out in 2022. Despite of over a century of research work, many intriguing problems on value distribution of exponential polynomials still remain unsolved. The role of exponential polynomials and their quotients in the theories of linear/non-linear differential equations, oscillation theory and differential-difference equations will also be discussed. Thirteen open problems are given to motivate the readers for further research in these topics.

Automated summary

This article introduces the definition and research history of exponential polynomials, discusses some unsolved problems, and explores the applications of exponential polynomials in various fields. Thirteen open problems are also provided to inspire further research.