# Exponential polynomials in the oscillation theory

## TL;DR

This paper investigates how exponential polynomials influence the oscillation of solutions to a second-order differential equation, highlighting the roles of specific polynomial components and geometric coefficients.

## Contribution

It introduces new oscillation criteria for differential equations with exponential polynomial coefficients, emphasizing the importance of the polynomial's leading terms and geometric properties.

## Key findings

- The polynomial's constant term affects solution oscillation.
- The geometric location of exponential coefficients influences oscillation behavior.
- Results include both entire plane and sectorial oscillation conditions.

## Abstract

Supposing that $A(z)$ is an exponential polynomial of the form $$ A(z)=H_0(z)+H_1(z)e^{\zeta_1z^n}+\cdots +H_m(z)e^{\zeta_mz^n}, $$ where $H_j$'s are entire and of order $<n$, it is demonstrated that the function $H_0(z)$ and the geometric location of the leading coefficients $\zeta_1,\ldots,\zeta_m$ play a key role in the oscillation of solutions of the differential equation $f''+A(z)f=0$. The key tools consist of value distribution properties of exponential polynomials, and elementary properties of the Phragm\'en-Lindel\"of indicator function. In addition to results in the whole complex plane, results on sectorial oscillation are proved.