On the rational solutions of generalized Abel equations
Abstract
We study the nonconstant rational solutions of generalized Abel equations x'=∑i=1s Ai(t)xni, s≥3, 1≤ n1<⋯<ns, with polynomial coefficients over ℝ or 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 ns−1 for the number of nonconstant rational solutions. Over ℝ, the same approach yields the additional bound min{ns−1,2s}. We construct a family of equations showing that both bounds are sharp for every s≥3. When n1=1, the real bound improves to min{ns−1,2(s−1)}. We then specialize to the three-term case with 1<n1<n2<n3, where we obtain a finer classification of the rational solutions. Under an additional antipodal separation condition, we obtain the real bound min{n3−1,5}, and show that the absolute constant $5$ is sharp.
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BibTeX
@misc{calderon2026rational,
title = {{On the rational solutions of generalized Abel equations}},
author = {L. A. Calderon and I. Ojeda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.10591},
eprint = {2605.10591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10591},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the rational solutions of generalized Abel equations
AU - L. A. Calderon
AU - I. Ojeda
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2605.10591
UR - https://arxiv.org/abs/2605.10591
AB - We study the nonconstant rational solutions of generalized Abel equations x'=∑_i=1^s A_i(t)x^n_i, s≥3, 1≤ n_1<⋯<n_s, with polynomial coefficients over ℝ or 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 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≥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.
ER -