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\n We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach builds upon a method proposed by Ferragut and Giacomini, whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series. We provide explicit bounds on the number of terms needed in the power series. This enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations. We then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity\n \n \n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n (<\/mml:mo>\n \n N<\/mml:mi>\n \n 2<\/mml:mn>\n \n \u03c9\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\tilde {\\mathcal {O}}(N^{2 \\omega })<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and a deterministic algorithm solving the problem in at most\n \n \n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n (<\/mml:mo>\n \n N<\/mml:mi>\n \n 2<\/mml:mn>\n \n \u03c9\n \n <\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\tilde {\\mathcal {O}}(N^{2 \\omega +1})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n arithmetic operations, where\u00a0\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denotes the given bound for the degree of the rational first integral, and\n \n \n \n \n \u03c9\n \n <\/mml:mi>\n \\omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in\n \n \n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n (<\/mml:mo>\n \n N<\/mml:mi>\n \n \n \u03c9\n \n <\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\tilde {\\mathcal {O}}(N^{\\omega +2})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n arithmetic operations. By comparison, the best previously known complexity was\n \n \n \n \n N<\/mml:mi>\n \n 4<\/mml:mn>\n \n \u03c9\n \n <\/mml:mi>\n +<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n N^{4\\omega +4}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package\n RationalFirstIntegrals<\/sc>\n which is available to interested readers with examples showing its efficiency.\n <\/p>","DOI":"10.1090\/mcom\/3007","type":"journal-article","created":{"date-parts":[[2015,8,3]],"date-time":"2015-08-03T10:13:23Z","timestamp":1438596803000},"page":"1393-1425","source":"Crossref","is-referenced-by-count":21,"title":["Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields"],"prefix":"10.1090","volume":"85","author":[{"given":"Alin","family":"Bostan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guillaume","family":"Ch\u00e8ze","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"Cluzeau","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jacques-Arthur","family":"Weil","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2015,8,3]]},"reference":[{"key":"1","isbn-type":"print","first-page":"51","article-title":"Translates of polynomials","author":"Abhyankar, Shreeram S.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/3764304442"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1145\/1073884.1073891","article-title":"Algebraic general solutions of algebraic ordinary differential equations","author":"Aroca, J. 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