rParametrizePlaneCurve -- Rational parametrization of rational plane curves.

Synopsis

Usage:

rParametrizePlaneCurve(I,J)
Inputs:
- I, an ideal, defining the plane curve C
- J, an ideal, the adjoint ideal of C.
Optional inputs:
- parametrizeConic => ..., default value null, Option whether to rationally parametrize conics.
Outputs:
- pI, a matrix, a column matrix with the rational parametrization of C.

Description

Computes a rational parametrization pI of C.

If the degree of C odd:

pI is over \mathbb{P}^{1}.

If the degree of C even:

pI is over a conic. So to get the conic apply "ideal ring" to the parametrization pI.

If the Option parametrizeConic=>true is given and C has a rational point then the conic is parametrized so pI is over \mathbb{P}^{1}.

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)

o5 = | t_0^2t_1t_2  |
     | -t_2^4       |
     | -t_0^4+t_2^4 |

            /QQ[t , t , t ]\       /QQ[t , t , t ]\
            |    0   1   2 |3      |    0   1   2 |1
o5 : Matrix |--------------|  <--- |--------------|
            |    2         |       |    2         |
            |   t  - t t   |       |   t  - t t   |
            \    1    0 2  /       \    1    0 2  /

i6 : rParametrizePlaneCurve(I,J,parametrizeConic=>true)

o6 = | t_0^3t_1^5  |
     | -t_0^8      |
     | t_0^8-t_1^8 |

                        3                  1
o6 : Matrix (QQ[t , t ])  <--- (QQ[t , t ])
                 0   1              0   1

i7 : K=QQ;

i8 : R=K[v,u,z];

i9 : I=ideal(u^5+2*u^2*v*z^2+2*u^3*v*z+u*v^2*z^2-4*u*v^3*z+2*v^5);

o9 : Ideal of R

i10 : J=ideal(u^3+v*u*z,v*u^2+v^2*z,v^2*u-u^2*z,v^3-v*u*z);

o10 : Ideal of R

i11 : rParametrizePlaneCurve(I,J)

o11 = | -2t_0^2t_1^3+t_0t_1^4 |
      | 4t_0^4t_1-2t_0^3t_1^2 |
      | -4t_0^5+t_1^5         |

                         3                  1
o11 : 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 pI will have entries in the homogeneous coordinate ring of \mathbb{P}^{1} over this extension.

Ways to use rParametrizePlaneCurve :

rParametrizePlaneCurve(Ideal,Ideal)

For the programmer

The object rParametrizePlaneCurve is a method function with options.