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dc.contributor.authorGiacomini, Héctor
dc.contributor.authorGiné, Jaume
dc.contributor.authorGrau Montaña, Maite
dc.date.accessioned2016-09-14T08:51:56Z
dc.date.available2016-09-14T08:51:56Z
dc.date.issued2007
dc.identifier.citationarXiv:math/0506036v1 [math.DS]
dc.identifier.issn0305-0041
dc.identifier.issn1469-8064
dc.identifier.urihttp://hdl.handle.net/10459.1/57802
dc.description.abstractWe study a planar polynomial differential system, given by . We consider a function , where gi(x) are algebraic functions of with ak(x) and algebraic functions, A0(x,y) and A1(x,y) do not share any common factor, h2(x) is a rational function, h(x) and h1(x) are functions of x with a rational logarithmic derivative and . We show that if I(x,y) is a first integral or an integrating factor, then I(x,y) is a Darboux function. A Darboux function is a function of the form , where fi and h are polynomials in and the λi's are complex numbers. In order to prove this result, we show that if g(x) is an algebraic particular solution, that is, if there exists an irreducible polynomial f(x,y) such that f(x,g(x)) ≡ 0, then f(x,y) = 0 is an invariant algebraic curve of the system. In relation with this fact, we give some characteristics related to particular solutions and functions of the form I(x,y) such as the structure of their cofactor. Moreover, we consider A0(x,y), A1(x,y) and h2(x) as before and a function of the form . We show that if the derivative of Φ(x,y) with respect to the flow is well defined over {(x,y): A0(x,y) = 0} then Φ(x,y) gives rise to an exponential factor. This exponential factor has the form exp {R(x,y)} where and with B1/B0 a function of the same form as h2A1/A0. Hence, exp {R(x,y)} factorizes as the product Φ(x,y) Ψ(x,y), for Ψ(x,y): = exp {B1/B0.ca_ES
dc.description.sponsorshipThe second and third authors are partially supported by a MCYT grant number BFM 2002- 04236-C02-01. The second author is also partially supported by DURSI of Government of Catalonia “Distinció de la Generalitat de Catalunya per a la promoció de la recerca universitària”.ca_ES
dc.language.isoengca_ES
dc.publisherCambridge University Pressca_ES
dc.relationMICYT/PN2000-2003/BFM2002-04236-C02-01ca_ES
dc.relation.isformatofVersió preprint del document publicat a https://doi.org/10.1017/S0305004107000497ca_ES
dc.relation.ispartofMathematical Proceedings of the Cambridge Philosophical Society, 2007, vol. 143, núm. 2, p. 487-508ca_ES
dc.relation.urihttp://arxiv.org/abs/math/0506036
dc.rights(c) Cambridge Philosophical Society 2007ca_ES
dc.subjectPlanar polynomial differential systemca_ES
dc.subjectAlgebraic functionca_ES
dc.subjectInvariant algebraic curveca_ES
dc.subjectIntegrabilityca_ES
dc.subject.otherEquacions diferencialsca_ES
dc.subject.otherÀlgebraca_ES
dc.titleThe role of algebraic solutions in planar polynomial differential systemsca_ES
dc.typearticleca_ES
dc.identifier.idgrec009820
dc.type.versionsubmittedVersionca_ES
dc.rights.accessRightsinfo:eu-repo/semantics/openAccessca_ES
dc.identifier.doihttps://doi.org/10.1017/S0305004107000497


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