Solving conics over function fields
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 595-606.

Soit F un corps de caractéristique différente de  2, et K=F(t). Nous donnons un algorithme simple pour trouver, étant donné a,b,cK * , une solution non-triviale dans K (si elle existe) à l’équation aX 2 +bY 2 +cZ 2 =0. Dans certains cas, l’algorithme a besoin d’une solution d’une équation similaire à coefficients dans F ; nous obtenons alors un algorithme récursif pour résoudre les coniques diagonales sur (t 1 ,,t n ) (en utilisant les algorithmes existants pour telles équations sur ) et sur 𝔽 q (t 1 ,,t n ).

Let F be a field whose characteristic is not 2 and K=F(t). We give a simple algorithm to find, given a,b,cK * , a nontrivial solution in K (if it exists) to the equation aX 2 +bY 2 +cZ 2 =0. The algorithm requires, in certain cases, the solution of a similar equation with coefficients in F; hence we obtain a recursive algorithm for solving diagonal conics over (t 1 ,,t n ) (using existing algorithms for such equations over ) and over 𝔽 q (t 1 ,,t n ).

@article{JTNB_2006__18_3_595_0,
     author = {van Hoeij, Mark and Cremona, John},
     title = {Solving conics over function fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {595--606},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {3},
     year = {2006},
     doi = {10.5802/jtnb.560},
     mrnumber = {2330429},
     zbl = {1129.11053},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.560/}
}
van Hoeij, Mark; Cremona, John. Solving conics over function fields. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 595-606. doi : 10.5802/jtnb.560. http://www.numdam.org/articles/10.5802/jtnb.560/

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