Complete solutions to a family of Thue equations of degree 12

Akinari Hoshi

doi:10.5802/jtnb.991

Complete solutions to a family of Thue equations of degree 12

Akinari Hoshi ¹

¹ Department of Mathematics Niigata University 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata 950-2181, Japan

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 549-568.

Résumé
Abstract

We considérons une famille paramétrique non galoisienne d’équations de Thue $F_{m} (x, y) = λ$ de degré $12$ où $m$ est un paramètre entier et où $λ$ est un diviseur de $729 (m^{2} + 3 m + 9)$ . En utilisant la méthode d’isomorphismes de corps développée dans [15], nous montrons que ces équations ont seulement des solutions triviales avec $x y (x + y) (x - y) (x + 2 y) (2 x + y) = 0$ .

We consider a parametric non-Galois family of Thue equations $F_{m} (x, y) = λ$ of degree $12$ where $m$ is an integral parameter and $λ$ is a divisor of $729 (m^{2} + 3 m + 9)$ . Using the field isomorphism method which is developed in [15], we show that the equations have only the trivial solutions with $x y (x + y) (x - y) (x + 2 y) (2 x + y) = 0$ .

Reçu le : 2015-12-29
Révisé le : 2016-07-04
Accepté le : 2016-09-11
Publié le : 2017-06-12

DOI : 10.5802/jtnb.991

Classification : 11D25, 11D41, 11R16, 11R20, 12F10
Mots clés : Thue equations, simplest cubic fields, simplest sextic fields.

Affiliations des auteurs :

Akinari Hoshi ¹

¹ Department of Mathematics Niigata University 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata 950-2181, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Complete solutions to a family of {Thue} equations of degree~12},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {549--568},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Akinari Hoshi. Complete solutions to a family of Thue equations of degree 12. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 549-568. doi : 10.5802/jtnb.991. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.991/

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