Solving conics over function fields

van Hoeij, Mark, and Cremona, John. "Solving conics over function fields." Journal de Théorie des Nombres de Bordeaux 18.3 (2006): 595-606. <http://eudml.org/doc/249634>.

@article{vanHoeij2006,
abstract = {Let $F$ be a field whose characteristic is not $2$ and $K = F(t)$. We give a simple algorithm to find, given $a,b,c \in K^*$, 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 $\mathbb\{Q\}(t_1,\dots ,t_n)$ (using existing algorithms for such equations over $\mathbb\{Q\}$) and over $\{\mathbb\{F\}\}_q(t_1,\dots ,t_n)$.},
affiliation = {Department of Mathematics Florida State University Tallahassee, FL 32306-3027, USA; School of Mathematical Sciences University of Nottingham University Park, Nottingham, NG7 2RD, UK},
author = {van Hoeij, Mark, Cremona, John},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {conic; solubility certificate; reduced term},
language = {eng},
number = {3},
pages = {595-606},
publisher = {Université Bordeaux 1},
title = {Solving conics over function fields},
url = {http://eudml.org/doc/249634},
volume = {18},
year = {2006},
}

TY - JOUR
AU - van Hoeij, Mark
AU - Cremona, John
TI - Solving conics over function fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 3
SP - 595
EP - 606
AB - Let $F$ be a field whose characteristic is not $2$ and $K = F(t)$. We give a simple algorithm to find, given $a,b,c \in K^*$, 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 $\mathbb{Q}(t_1,\dots ,t_n)$ (using existing algorithms for such equations over $\mathbb{Q}$) and over ${\mathbb{F}}_q(t_1,\dots ,t_n)$.
LA - eng
KW - conic; solubility certificate; reduced term
UR - http://eudml.org/doc/249634
ER -