{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T03:49:57Z","timestamp":1776743397766,"version":"3.51.2"},"reference-count":16,"publisher":"World Scientific Pub Co Pte Lt","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2012,11]]},"abstract":"<jats:p> Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in \u211d<jats:sup>n<\/jats:sup> of degree m is at least [Formula: see text], then the vector field has a rational first integral. Llibre and Zhang used only Linear Algebra to provide a shorter and easier proof of the result given by Jouanolou. We use ideas of Llibre and Zhang to extend the Jouanolou result to polynomial vector fields defined on algebraic regular hypersurfaces of \u211d<jats:sup>n+1<\/jats:sup>, this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of \u211d<jats:sup>n+1<\/jats:sup>. <\/jats:p>","DOI":"10.1142\/s0218127412502707","type":"journal-article","created":{"date-parts":[[2012,12,13]],"date-time":"2012-12-13T09:14:17Z","timestamp":1355390057000},"page":"1250270","source":"Crossref","is-referenced-by-count":9,"title":["RATIONAL FIRST INTEGRALS FOR POLYNOMIAL VECTOR FIELDS ON ALGEBRAIC HYPERSURFACES OF \u211d<sup>n+1<\/sup>"],"prefix":"10.1142","volume":"22","author":[{"given":"JAUME","family":"LLIBRE","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain"}]},{"given":"YUDY","family":"BOLA\u00d1OS","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain"}]}],"member":"219","published-online":{"date-parts":[[2012,12,13]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(81)90644-7"},{"key":"rf2","first-page":"268","volume":"128","author":"Bountis T. C.","journal-title":"J. Phys. A"},{"key":"rf3","first-page":"467","volume":"23","author":"Cantrijn F.","journal-title":"SIAM Rev."},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/BF02546499"},{"key":"rf5","first-page":"1012","volume":"86","author":"Darboux G.","journal-title":"C. R. Math. Acad. Sci. Par\u00eds"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/24\/19\/020"},{"key":"rf7","doi-asserted-by":"crossref","first-page":"159","DOI":"10.24033\/asens.295","volume":"4","author":"Goursat E.","journal-title":"Ann. Sci. \u00c9cole Norm. Sup."},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0370-1573(87)90089-5"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1070\/RM1976v031n05ABEH004190"},{"key":"rf10","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0063393","volume-title":"Equations de Plaff Alg\u00e9briques","volume":"708","author":"Jouanolou J. P.","year":"1979"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160210503"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02871455"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2007.12.001"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-78-04537-4"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0274-2"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(88)91018-3"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127412502707","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T18:50:04Z","timestamp":1565117404000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127412502707"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11]]},"references-count":16,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2012,12,13]]},"published-print":{"date-parts":[[2012,11]]}},"alternative-id":["10.1142\/S0218127412502707"],"URL":"https:\/\/doi.org\/10.1142\/s0218127412502707","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,11]]}}}