Function: qfparam
Section: linear_algebra
C-Name: qfparam
Prototype: GGD0,L,
Help: qfparam(G, sol, {flag = 0}):
coefficients of binary quadratic forms that parametrize the
solutions of the ternary quadratic form G, using the particular
solution sol.
Doc: coefficients of binary quadratic forms that parametrize the
solutions of the ternary quadratic form $G$, using the particular
solution~\var{sol}.
\fl is optional and can be 1, 2, or 3, in which case the \fl-th form is
reduced. The default is \fl=0 (no reduction).
\bprog
? G = [1,0,0;0,1,0;0,0,-34];
? M = qfparam(G, qfsolve(G))
%2 =
[ 3 -10 -3]
[-5 -6 5]
[ 1 0 1]
@eprog
Indeed, the solutions can be parametrized as
$$(3x^2 - 10xy - 3y^2)^2 + (-5x^2 - 6xy + 5y^2)^2 -34(x^2 + y^2)^2 = 0.$$
\bprog
? v = y^2 * M*[1,x/y,(x/y)^2]~
%3 = [3*x^2 - 10*y*x - 3*y^2, -5*x^2 - 6*y*x + 5*y^2, -x^2 - y^2]~
? v~*G*v
%4 = 0
@eprog