On the rational solutions of generalized Abel equations

Abstract

We study the nonconstant rational solutions of generalized Abel equations \[ x'=\sum_{i=1}^s A_i(t)x^{n_i},\qquad s\geq3,\qquad 1\leq n_1<\cdots<n_s, \] with polynomial coefficients over $\mathbb{R}$ or $\mathbb{C}$. Using the Newton diagram at infinity and the associated edge polynomials, we obtain, under simplicity and nonresonance conditions on the edge polynomials, the global upper bound $n_s-1$ for the number of nonconstant rational solutions. Over $\mathbb{R}$, the same approach yields the additional bound $\min \{n_s-1,2s\}$. We construct a family of equations showing that both bounds are sharp for every $s\ge 3$. When $n_1=1$, the real bound improves to $\min \{n_s-1,2(s-1)\}$. We then specialize to the three-term case with $1<n_1<n_2<n_3$, where we obtain a finer classification of the rational solutions. Under an additional antipodal separation condition, we obtain the real bound $\min \{n_3-1,5\}$, and show that the absolute constant $5$ is sharp.