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rParametrizeConic -- Compute a rational parametrization of a conic.

Synopsis

Usage:
rParametrizeConic(I)
Inputs:
- I, an ideal, of an irreducible conic.
Outputs:
- a matrix, with the rational parametrization, or the 0-matrix if the conic does not have a rational point.

Description

Computes a rational parametrization of a conic or returns the 0-matrix if the conic does not have a rational point.

i1 : K=QQ;

i2 : R=K[y_0..y_2];

i3 : I=ideal(y_0*y_2-y_1^2+y_2^2);

o3 : Ideal of R

i4 : rParametrizeConic(I)

o4 = | t_0^2-t_1^2 |
     | -t_0t_1     |
     | -t_0^2      |

                        3                  1
o4 : Matrix (QQ[t , t ])  <--- (QQ[t , t ])
                 0   1              0   1

Caveat

If the conic does not have a rational point the field extension of degree 2 should be implemented.

Ways to use `rParametrizeConic` :

rParametrizeConic(Ideal)