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Volume 18, Issue 3

# Polynomial Solutions of Equivariant Polynomial Abel Differential Equations

Jaume Llibre
• Corresponding author
• Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
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/ Clàudia Valls
• Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal
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Published Online: 2017-12-16 | DOI: https://doi.org/10.1515/ans-2017-6043

## Abstract

Let $a\left(x\right)$ be non-constant and let ${b}_{j}\left(x\right)$, for $j=0,1,2,3$, be real or complex polynomials in the variable x. Then the real or complex equivariant polynomial Abel differential equation $a\left(x\right)\stackrel{˙}{y}={b}_{1}\left(x\right)y+{b}_{3}\left(x\right){y}^{3}$, with ${b}_{3}\left(x\right)\ne 0$, and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind $a\left(x\right)y\stackrel{˙}{y}={b}_{0}\left(x\right)+{b}_{2}\left(x\right){y}^{2}$, with ${b}_{2}\left(x\right)\ne 0$, have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.

MSC 2010: 34A05; 34C05; 37C10

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Revised: 2017-11-10

Accepted: 2017-11-27

Published Online: 2017-12-16

Published in Print: 2018-08-01

The first author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013.

Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 3, Pages 537–542, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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