{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:15:51Z","timestamp":1776802551200,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"308","license":[{"start":{"date-parts":[[2018,4,7]],"date-time":"2018-04-07T00:00:00Z","timestamp":1523059200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00c3\u00ada y Competitividad","doi-asserted-by":"publisher","award":["MTM2014-54141-P"],"award-info":[{"award-number":["MTM2014-54141-P"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {K}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a characteristic zero field, let\n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \\alpha<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be an algebraic element over\n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {K}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {C}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n a rational curve defined over\n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {K}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n given by a parametrization\n \n \n \n \n \u03c8\n \n <\/mml:mi>\n \\psi<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with coefficients in\n \n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \u03b1\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathbb {K}(\\alpha )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We propose an algorithm to solve the following problem, that is, a parametric version of Hilbert-Hurwitz: To compute a linear fraction\n \n \n \n \n u<\/mml:mi>\n =<\/mml:mo>\n \n \n a<\/mml:mi>\n t<\/mml:mi>\n +<\/mml:mo>\n b<\/mml:mi>\n <\/mml:mrow>\n \n c<\/mml:mi>\n t<\/mml:mi>\n +<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n u=\\frac {at+b}{ct+d}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n \n \u03c8\n \n <\/mml:mi>\n (<\/mml:mo>\n u<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\psi (u)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has coefficients over an algebraic extension of\n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {K}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of degree at most two and a conic\n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {K}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -birational to\n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {C}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Moreover, if the degree of\n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {C}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is odd or\n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \\alpha<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is of odd degree over\n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {K}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we can compute a parametrization of\n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {C}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with coefficients over\n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {K}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The problem is solved without implicitization methods nor analyzing the singularities\u00a0of\u00a0\n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {C}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/mcom\/3202","type":"journal-article","created":{"date-parts":[[2016,8,17]],"date-time":"2016-08-17T09:40:53Z","timestamp":1471426853000},"page":"3001-3018","source":"Crossref","is-referenced-by-count":0,"title":["A parametric version of the Hilbert-Hurwitz theorem using hypercircles"],"prefix":"10.1090","volume":"86","author":[{"given":"Luis Felipe","family":"Tabera","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2017,4,7]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1145\/258726.258844","article-title":"A relatively optimal rational space curve reparametrization algorithm through canonical divisors","author":"Andradas, Carlos","year":"1997"},{"key":"2","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1145\/309831.309845","article-title":"Base field restriction techniques for parametric curves","author":"Andradas, Carlos","year":"1999"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1016\/j.jsc.2008.09.001","article-title":"On the simplification of the coefficients of a parametrization","volume":"44","author":"Andradas, Carlos","year":"2009","journal-title":"J. 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Villarino, Algoritmos de optimalidad algebraica y de cuasi-polinomialidad para curvas racionales, Ph.D. thesis, Universidad de Alcal\u00e1, 2007."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-308\/S0025-5718-2017-03202-6\/mcom3202_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2017-86-308\/S0025-5718-2017-03202-6\/S0025-5718-2017-03202-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-308\/S0025-5718-2017-03202-6\/S0025-5718-2017-03202-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:22:44Z","timestamp":1776799364000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-308\/S0025-5718-2017-03202-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,7]]},"references-count":25,"journal-issue":{"issue":"308","published-print":{"date-parts":[[2017,11]]}},"alternative-id":["S0025-5718-2017-03202-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3202","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2017,4,7]]}}}