Exponential polynomials in the oscillation theory

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.