# 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:
• Outputs:
• pI, , 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 ℙ1.

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 ℙ1.

 `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 ℙ1 over this extension.