dc.contributor.author Giacomini, Héctor dc.contributor.author Giné, Jaume dc.contributor.author Grau Montaña, Maite dc.date.accessioned 2016-09-14T08:51:56Z dc.date.available 2016-09-14T08:51:56Z dc.date.issued 2007 dc.identifier.citation arXiv:math/0506036v1 [math.DS] dc.identifier.issn 0305-0041 dc.identifier.issn 1469-8064 dc.identifier.uri http://hdl.handle.net/10459.1/57802 dc.description.abstract We 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. ca_ES 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. dc.description.sponsorship The 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.iso eng ca_ES dc.publisher Cambridge University Press ca_ES dc.relation MICYT/PN2000-2003/BFM2002-04236-C02-01 ca_ES dc.relation.isformatof Versió preprint del document publicat a https://doi.org/10.1017/S0305004107000497 ca_ES dc.relation.ispartof Mathematical Proceedings of the Cambridge Philosophical Society, 2007, vol. 143, núm. 2, p. 487-508 ca_ES dc.relation.uri http://arxiv.org/abs/math/0506036 dc.rights (c) Cambridge Philosophical Society 2007 ca_ES dc.subject Planar polynomial differential system ca_ES dc.subject Algebraic function ca_ES dc.subject Invariant algebraic curve ca_ES dc.subject Integrability ca_ES dc.subject.other Equacions diferencials ca_ES dc.subject.other Àlgebra ca_ES dc.title The role of algebraic solutions in planar polynomial differential systems ca_ES dc.type article ca_ES dc.identifier.idgrec 009820 dc.type.version submittedVersion ca_ES dc.rights.accessRights info:eu-repo/semantics/openAccess ca_ES dc.identifier.doi https://doi.org/10.1017/S0305004107000497
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