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parametrizeConic -- Option whether to rationally parametrize conics.

Description

If this option is set true and the computation of rParametrizePlaneCurve or parametrize leads to a conic (even degree case) and this conic has a rational point then this conic is also parametrized. So the final result will be a parametrization over \mathbb{P}^{1}. If the conic does not have a rational point a warning is displayed and the parametrization over the conic is returned.

i1 : K=QQ;
 
i2 : R=K[v,u,z];
 
i3 : I=ideal(v^8-u^3*(z+u)^5);

o3 : Ideal of R
 
i4 : J=ideal(u^6+4*u^5*z+6*u^4*z^2+4*u^3*z^3+u^2*z^4,v*u^5+3*v*u^4*z+3*v*u^3*z^2+v*u^2*z^3,v^2*u^4+3*v^2*u^3*z+3*v^2*u^2*z^2+v^2*u*z^3,v^3*u^3+2*v^3*u^2*z+v^3*u*z^2,v^4*u^2+v^4*u*z,v^5*u+v^5*z,v^6);

o4 : Ideal of R
 
i5 : rParametrizePlaneCurve(I,J,parametrizeConic=>true)

o5 = | t_0^3t_1^5  |
     | -t_0^8      |
     | t_0^8-t_1^8 |

                        3                  1
o5 : Matrix (QQ[t , t ])  <--- (QQ[t , t ])
                 0   1              0   1

Caveat

If rParametrizeConic is changed such that it passes to a degree 2 field extension if the degree of C is even and the conic does not have a rational point, then the result will have entries in the homogeneous coordinate ring of \mathbb{P}^{1} over this extension.

See also

Functions with optional argument named parametrizeConic :

For the programmer

The object parametrizeConic is a symbol.