Rational Solutions of Abel Differential Equations

TL;DR

This paper investigates the conditions under which Abel differential equations have rational solutions, establishing bounds on their number and linking solutions to Darboux integrability.

Contribution

It provides new bounds on the number of rational solutions for Abel equations based on polynomial degrees and connects solution counts to Darboux first integrals.

Findings

At most two rational solutions when deg(A) is even or deg(B) exceeds a certain threshold.

Upper bounds on rational solutions in other cases.

Existence of Darboux first integral when solutions exceed a certain number.

Abstract

We study the rational solutions of the Abel equation $x'=A(t)x^3+B(t)x^2$ where $A,B\in C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$ then the equation has at most two rational solutions. For any other case, an upper bound on the number of rational solutions is obtained. Moreover, we prove that if there are more than $(deg(A)+1)/2$ rational solutions then the equation admits a Darboux first integral.