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 $A_i\in\Bbbk[t]$, $\Bbbk\in\{\mathbb R,\mathbb C\}$. 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.