{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T13:27:51Z","timestamp":1772285271260,"version":"3.50.1"},"reference-count":5,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,2,1]],"date-time":"2010-02-01T00:00:00Z","timestamp":1264982400000},"content-version":"unspecified","delay-in-days":3318,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2001]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper is concerned with a relation of Darboux in enumerative geometry, which has very useful applications in the study of polynomial vector fields. The original statement of Darboux was not correct. The present paper gives two different elementary proofs of this relation. The first one follows the ideas of Darboux, and uses basic facts about the intersection index of two plane algebraic curves; the second proof is rather more sophisticated, and gives a stronger result, which should also be very useful. The power of the relation of Darboux is then illustrated by the provision of new, simple proofs of two known results. First, it is shown that an irreducible invariant algebraic curve of degree <jats:italic>n<\/jats:italic> &gt; 1 without multiple points for a polynomial vector field of degree <jats:italic>m<\/jats:italic> satisfies <jats:italic>n<\/jats:italic> \u2264 <jats:italic>m<\/jats:italic> + 1. Second, a proof is given that quadratic polynomial vector fields have no algebraic limit cycles of degree 3.<\/jats:p>","DOI":"10.1112\/s1461157000000863","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T11:42:44Z","timestamp":1375789364000},"page":"197-210","source":"Crossref","is-referenced-by-count":18,"title":["On a Result of Darboux"],"prefix":"10.1112","volume":"4","author":[{"given":"Javier","family":"Chavarriga","sequence":"first","affiliation":[]},{"given":"Jaume","family":"Llibre","sequence":"additional","affiliation":[]},{"given":"Jean Moulin","family":"Ollagnier","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2010,2,1]]},"reference":[{"key":"S1461157000000863_ref005","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/34\/3\/325"},{"key":"S1461157000000863_ref001","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.2000.3891"},{"key":"S1461157000000863_ref002","first-page":"60","article-title":"\u2018M\u00e9moire sur les \u00e9quations diff\u00e9rentielles alg\u00e9briques du premier ordre et du premier degr\u00e9\u2019","volume":"2","author":"Darboux","year":"1878","journal-title":"Bull. Sci. Math."},{"key":"S1461157000000863_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0063393"},{"key":"S1461157000000863_ref003","volume-title":"Algebraic curves, Math. Lecture Ser.","author":"Fulton","year":"1969"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157000000863","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,6]],"date-time":"2019-06-06T18:28:50Z","timestamp":1559845730000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157000000863\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001]]},"references-count":5,"alternative-id":["S1461157000000863"],"URL":"https:\/\/doi.org\/10.1112\/s1461157000000863","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001]]}}}