Solving conics over function fields

van Hoeij, Mark; Cremona, John

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 $K=F\left(t\right)$. Nous donnons un algorithme simple pour trouver, étant donné $a,b,c\in K^{*}$, une solution non-triviale dans $K$ (si elle existe) à l’équation $a{X}^2+b{Y}^2+c{Z}^2=0$. 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 $\mathbb{Q}(t_1,\cdots ,t_n)$ (en utilisant les algorithmes existants pour telles équations sur $\mathbb{Q}$) et sur ${𝔽}_q(t_1,\cdots ,t_n)$.

Let $F$ be a field whose characteristic is not $2$ and $K=F\left(t\right)$. We give a simple algorithm to find, given $a,b,c\in K^{*}$, a nontrivial solution in $K$ (if it exists) to the equation $a{X}^2+b{Y}^2+c{Z}^2=0$. 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 $\mathbb{Q}(t_1,\cdots ,t_n)$ (using existing algorithms for such equations over $\mathbb{Q}$) and over ${𝔽}_q(t_1,\cdots ,t_n)$.

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