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Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


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Volume 18, Issue 3

Issues

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(x) be non-constant and let bj(x), 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(x)y˙=b1(x)y+b3(x)y3, with b3(x)0, and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind a(x)yy˙=b0(x)+b2(x)y2, with b2(x)0, have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.

Keywords: Polynomial Abel Equations; Equivariant Polynomial Equation; Polynomial Solutions

MSC 2010: 34A05; 34C05; 37C10

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About the article


Received: 2017-04-28

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, DOI: https://doi.org/10.1515/ans-2017-6043.

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