L. A. Calderon | Searcharxiv

arXiv2026

On the rational solutions of generalized Abel equations

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.

math.CA↗

arXiv2021

Rational Solutions of Abel Differential Equations

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.

math.CA↗