On global non-oscillation of linear ordinary differential equations with polynomial coefficients

TL;DR

This paper characterizes when linear ODEs with polynomial coefficients are globally non-oscillating on the complex projective line, linking it to Fuchsian equations and the nature of characteristic exponents, and provides bounds on zeros of exponential polynomials.

Contribution

It establishes a precise criterion for global non-oscillation of polynomial coefficient ODEs and derives a new explicit upper bound for zeros of exponential polynomials.

Findings

Linear ODEs with polynomial coefficients are globally non-oscillating iff they are Fuchsian.

At each singular point, characteristic exponents have distinct real parts.

New explicit upper bound for zeros of exponential polynomials in a strip.

Abstract

In this note we show that a linear ordinary differential equation with polynomial coefficients is globally non-oscillating in $\mathbb{C} P^1$ if and only if it is Fuchsian, and at every its singular point any two distinct characteristic exponents have distinct real parts. As a byproduct of our study, we obtain a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip.