Solving conics over function fields
John Cremona
University of Nottingham
(Joint work with Mark van Hoeij.) 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
$\Q(t_1,\dots,t_n)$ (using existing algorithms for such equations
over~$\Q$) and over $\F_q(t_1,\dots,t_n)$.