On the rational solutions of generalized Abel equations

Abstract

We study nonconstant rational solutions of \[ x'=A_3(t)x^{n_3}+A_2(t)x^{n_2}+A_1(t)x^{n_1}, \qquad 1<n_1<n_2<n_3, \] with $A_i\in \mathbb{k}[t]$, $\mathbb{k}\in \{\mathbb{R},\mathbb{C}\}$. We prove that every such solution is of the form $x=1/p(t)$, and use the Newton--Puiseux polygon at infinity to restrict the possible degrees of $p$. Under a nondegeneracy hypothesis, the associated edge polynomials yield explicit bounds for the total number $\mathcal{S}$ of rational solutions. In particular, $\mathcal{S}\le (n_2-1)+2(n_3-1)$ over $\mathbb{C}$, while over $\mathbb{R}$ one has $\mathcal{S}\le 12$, with sharper parity-dependent estimates in the real case.