{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T10:07:11Z","timestamp":1760954831393},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7316,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,3]]},"abstract":"<jats:p>Nous dirons qu'un corps (commutatif) <jats:italic>K<\/jats:italic> est un <jats:italic>corps de courbe r\u00e9elle<\/jats:italic> sur le corps <jats:italic>k<\/jats:italic> si et seulement si c'est une extension de <jats:italic>k<\/jats:italic>, ordonnable, finiment engendr\u00e9e et de degr\u00e9 de transcendance 1 sur <jats:italic>k<\/jats:italic>, telle que <jats:italic>k<\/jats:italic> soit relativement alg\u00e9briquement clos dans <jats:italic>K<\/jats:italic>. Un tel corps est le corps d'une courbe (plane) d\u00e9finie sur <jats:italic>k<\/jats:italic>. De plus si <jats:italic>k<\/jats:italic> est r\u00e9el-clos, toute courbe d\u00e9finie sur <jats:italic>k<\/jats:italic> dont <jats:italic>K<\/jats:italic> est le corps, a un point r\u00e9gulier rationnel sur <jats:italic>k<\/jats:italic>; en effet soit <jats:italic>I<\/jats:italic> \u2282 <jats:italic>k<\/jats:italic>[<jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>,\u2026, <jats:italic>X<jats:sub>n<\/jats:sub><\/jats:italic>] l'id\u00e9al d'une telle courbe \u0393; on a donc: <jats:italic>K<\/jats:italic> = <jats:italic>k<\/jats:italic>[<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>,\u2026, <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic>] o\u00f9 <jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>,\u2026, <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic> sont les classes d'\u00e9quivalence de <jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>,\u2026, <jats:italic>X<jats:sub>n<\/jats:sub><\/jats:italic>; (<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>,\u2026, <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic>) est <jats:italic>un<\/jats:italic> point r\u00e9gulier de \u0393 dans la cl\u00f4ture r\u00e9elle de <jats:italic>K<\/jats:italic>, et par mod\u00e8le-compl\u00e9tude, \u0393 a un point r\u00e9gulier rationnel sur <jats:italic>k<\/jats:italic>.<\/jats:p><jats:p>Notre but est d'\u00e9tendre les r\u00e9sultats de Raphael Robinson [6] sur l'ind\u00e9cidabilit\u00e9 des corps de fractions rationnelles sur un corps ordonnable (dans le langage des corps). Comme lui, nous donnerons une m\u00e9thode qui ne s'applique que lorsque le corps de base <jats:italic>k<\/jats:italic> est r\u00e9el-clos, et une seconde m\u00e9thode g\u00e9n\u00e9rate.<\/jats:p>","DOI":"10.2307\/2275251","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:51:08Z","timestamp":1146941468000},"page":"87-91","source":"Crossref","is-referenced-by-count":3,"title":["Ind\u00e9cidabilit\u00e9 des corps de courbe r\u00e9elle"],"prefix":"10.1017","volume":"59","author":[{"given":"Luc","family":"B\u00e9lair","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Louis","family":"Duret","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200020247_ref004","volume-title":"Vari\u00e9t\u00e9s ab\u00e9liennes","author":"N\u00e9ron"},{"key":"S0022481200020247_ref003","volume-title":"Elliptic functions","author":"Lang"},{"key":"S0022481200020247_ref006","first-page":"275","volume-title":"Zeitschrift f\u00fcr Matematische Logik und Grundlagen der Mathematik, vol. 10","author":"Robinson","year":"1964"},{"key":"S0022481200020247_ref002","volume-title":"Algebraic curves","author":"Fulton"},{"key":"S0022481200020247_ref001","first-page":"948","volume":"51","author":"Duret","year":"1968","journal-title":"Sur la th\u00e9orie \u00e9l\u00e9mentaire des corps de fonctions"},{"key":"S0022481200020247_ref005","volume-title":"Elliptic curves","volume":"326","author":"Robert"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200020247","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T15:17:00Z","timestamp":1557933420000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200020247\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,3]]},"references-count":6,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1994,3]]}},"alternative-id":["S0022481200020247"],"URL":"https:\/\/doi.org\/10.2307\/2275251","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,3]]}}}