On the rational solutions of generalized Abel equations

TL;DR

This paper characterizes rational solutions of generalized Abel equations, showing they are reciprocals of polynomials and providing bounds on their total number using algebraic and geometric methods.

Contribution

It proves all rational solutions are of the form 1/p(t) and derives explicit bounds on their number using Newton--Puiseux polygons and edge polynomials.

Findings

All rational solutions are of the form 1/p(t).

Explicit bounds on the number of solutions over complex and real fields.

Use of Newton--Puiseux polygons to restrict degrees of solutions.

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.