Darboux theory of integrability for polynomial vector fields in Rn taking into account the multiplicity at infinity Export

@article{citeulike:5347702, abstract = {Darboux theory of integrability was established by Darboux in 1878, which provided a relation between the existence of first integrals and invariant algebraic hypersurfaces of vector fields in inlMMLBox or inlMMLBox with n 2 . Jouanolou 1979 improved this theory to obtain rational first integrals via invariant algebraic surfaces using sophisticated tools of algebraic geometry. Recently in [J. Llibre, X. Zhang, Darboux theory of integrability in inlMMLBox taking into account the multiplicity, J. Differential Equations, in press] this theory was improved taking into account not only the invariant algebraic hypersurfaces but also their multiplicity. In this paper we will show that if the hyperplane at infinity for a polynomial vector field in inlMMLBox has multiplicity larger than 1, we can improve again the Darboux theory of integrability. We also show some difficulties for obtaining an extension of this result to polynomial vector fields in inlMMLBox .}, author = {Llibre, Jaume and Zhang, Xiang}, citeulike-article-id = {5347702}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.bulsci.2009.06.002}, citeulike-linkout-1 = {http://linkinghub.elsevier.com/retrieve/pii/S0007449709000396}, day = {26}, doi = {10.1016/j.bulsci.2009.06.002}, issn = {00074497}, journal = {Bulletin des Sciences Math\'{e}matiques}, keywords = {algebraic\_geometry, complex\_analysis, integrability, polynomial\_vector\_fields}, month = {June}, posted-at = {2009-10-28 07:28:12}, priority = {4}, title = {Darboux theory of integrability for polynomial vector fields in Rn taking into account the multiplicity at infinity}, url = {http://dx.doi.org/10.1016/j.bulsci.2009.06.002}, year = {2009} }

TY - JOUR ID - citeulike:5347702 L3 - citeulike-article-id:5347702 N2 - Darboux theory of integrability was established by Darboux in 1878, which provided a relation between the existence of first integrals and invariant algebraic hypersurfaces of vector fields in inlMMLBox or inlMMLBox with n 2 . Jouanolou 1979 improved this theory to obtain rational first integrals via invariant algebraic surfaces using sophisticated tools of algebraic geometry. Recently in [J. Llibre, X. Zhang, Darboux theory of integrability in inlMMLBox taking into account the multiplicity, J. Differential Equations, in press] this theory was improved taking into account not only the invariant algebraic hypersurfaces but also their multiplicity. In this paper we will show that if the hyperplane at infinity for a polynomial vector field in inlMMLBox has multiplicity larger than 1, we can improve again the Darboux theory of integrability. We also show some difficulties for obtaining an extension of this result to polynomial vector fields in inlMMLBox . JF - Bulletin des Sciences Mathématiques SN - 00074497 TI - Darboux theory of integrability for polynomial vector fields in Rn taking into account the multiplicity at infinity KW - algebraic_geometry KW - complex_analysis KW - integrability KW - polynomial_vector_fields AU - Llibre, Jaume AU - Zhang, Xiang PY - 2009/06/26/ UR - http://dx.doi.org/10.1016/j.bulsci.2009.06.002 ER -