Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials

In 1891 Poincaré asked: Given m ≥ 2, is there a positive integer M(m) such that if a polynomial differential system of degree m has an invariant algebraic curve of degree ≥ M(m), then it has a rational first integral? Brunella and Mendes repeated the same open question in 2000, and Lins-Neto in 2002...

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Autores: Llibre, Jaume|||0000-0002-9511-5999, Valls, Clàudia|||0000-0001-8279-1229
Tipo de recurso: artículo
Fecha de publicación:2025
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:318295
Acceso en línea:https://ddd.uab.cat/record/318295
https://dx.doi.org/urn:doi:10.1016/j.jpaa.2025.108001
Access Level:acceso embargado
Palabra clave:Polynomial differential systems
Invariant algebraic curve
Rational first integral
Hermite polynomials
Laguerre polynomials
Legendre polynomials
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spelling Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomialsLlibre, Jaume|||0000-0002-9511-5999Valls, Clàudia|||0000-0001-8279-1229Polynomial differential systemsInvariant algebraic curveRational first integralHermite polynomialsLaguerre polynomialsLegendre polynomialsIn 1891 Poincaré asked: Given m ≥ 2, is there a positive integer M(m) such that if a polynomial differential system of degree m has an invariant algebraic curve of degree ≥ M(m), then it has a rational first integral? Brunella and Mendes repeated the same open question in 2000, and Lins-Neto in 2002. Between the years 2001 and 2003 three different families of quadratic polynomial differential systems provided a negative answer to this question. One of the answers used the Hermite polynomials. Recently a new negative answer was provided for polynomial differential systems of arbitrary degree using the Laguerre polynomials. In this paper we provide another new negative answer but using for first time the Legendre polynomials. So the orthogonal polynomials play a role in the Poincaré's question. Moreover we classify the phase portraits of these polynomial differential systems having invariant algebraic curves of arbitrary degree based on the Legendre polynomials. 220252025-01-0120272027-08-31Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articlehttps://ddd.uab.cat/record/318295https://dx.doi.org/urn:doi:10.1016/j.jpaa.2025.108001reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2022-136613NB-I00Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00113embargoed accesshttp://purl.org/coar/access_right/c_f1cfAquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.https://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/embargoedAccessoai:ddd.uab.cat:3182952026-06-06T12:50:31Z
dc.title.none.fl_str_mv Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials
title Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials
spellingShingle Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials
Llibre, Jaume|||0000-0002-9511-5999
Polynomial differential systems
Invariant algebraic curve
Rational first integral
Hermite polynomials
Laguerre polynomials
Legendre polynomials
title_short Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials
title_full Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials
title_fullStr Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials
title_full_unstemmed Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials
title_sort Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials
dc.creator.none.fl_str_mv Llibre, Jaume|||0000-0002-9511-5999
Valls, Clàudia|||0000-0001-8279-1229
author Llibre, Jaume|||0000-0002-9511-5999
author_facet Llibre, Jaume|||0000-0002-9511-5999
Valls, Clàudia|||0000-0001-8279-1229
author_role author
author2 Valls, Clàudia|||0000-0001-8279-1229
author2_role author
dc.subject.none.fl_str_mv Polynomial differential systems
Invariant algebraic curve
Rational first integral
Hermite polynomials
Laguerre polynomials
Legendre polynomials
topic Polynomial differential systems
Invariant algebraic curve
Rational first integral
Hermite polynomials
Laguerre polynomials
Legendre polynomials
description In 1891 Poincaré asked: Given m ≥ 2, is there a positive integer M(m) such that if a polynomial differential system of degree m has an invariant algebraic curve of degree ≥ M(m), then it has a rational first integral? Brunella and Mendes repeated the same open question in 2000, and Lins-Neto in 2002. Between the years 2001 and 2003 three different families of quadratic polynomial differential systems provided a negative answer to this question. One of the answers used the Hermite polynomials. Recently a new negative answer was provided for polynomial differential systems of arbitrary degree using the Laguerre polynomials. In this paper we provide another new negative answer but using for first time the Legendre polynomials. So the orthogonal polynomials play a role in the Poincaré's question. Moreover we classify the phase portraits of these polynomial differential systems having invariant algebraic curves of arbitrary degree based on the Legendre polynomials.
publishDate 2025
dc.date.none.fl_str_mv
2
2025
2025-01-01
2027
2027-08-31
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AM
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dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
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dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/318295
https://dx.doi.org/urn:doi:10.1016/j.jpaa.2025.108001
url https://ddd.uab.cat/record/318295
https://dx.doi.org/urn:doi:10.1016/j.jpaa.2025.108001
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2022-136613NB-I00
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00113
dc.rights.none.fl_str_mv embargoed access
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https://creativecommons.org/licenses/by-nc-nd/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/embargoedAccess
rights_invalid_str_mv embargoed access
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https://creativecommons.org/licenses/by-nc-nd/4.0/
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dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
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collection Dipòsit Digital de Documents de la UAB
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