parametrize -- Rational parametrization of rational curves.

Synopsis

• Usage:
parametrize(I)
parametrize(I,J)
• Inputs:
• I, an ideal, defining a rational plane curve C or rational normal curve
• J, an ideal, the adjoint ideal of the plane curve C.
• Optional inputs:
• parametrizeConic => ..., default value null, Option whether to rationally parametrize conics.
• 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 \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 hence pI is over \mathbb{P}^{1}.

If the second argument J is not specified and degree of C is bigger than 2 then J is being computed via the package AdjointIdeal.

If the function is applied to a rational normal curve it calls rParametrizeRNC.

If it is applied to a plane conic it calls rParametrizeConic.

 i1 : K=QQ; i2 : R=K[v,u,z]; i3 : I=ideal(v^8-u^3*(z+u)^5); o3 : Ideal of R i4 : p=parametrize(I) o4 = | 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 o4 : Matrix |--------------| <--- |--------------| | 2 | | 2 | | t - t t | | t - t t | \ 1 0 2 / \ 1 0 2 / i5 : parametrize(I,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 i6 : Irnc=mapToRNC(I); o6 : Ideal of QQ[x , x , x , x , x , x , x ] 0 1 2 3 4 5 6 i7 : parametrize(Irnc) o7 = | t_0^2t_2 | | -t_1t_2^2 | | t_0^3 | | -t_0t_1t_2 | | t_2^3 | | -t_0^2t_1 | | t_0t_2^2 | /QQ[t , t , t ]\ /QQ[t , t , t ]\ | 0 1 2 |7 | 0 1 2 |1 o7 : Matrix |--------------| <--- |--------------| | 2 | | 2 | | t - t t | | t - t t | \ 1 0 2 / \ 1 0 2 / i8 : parametrize(Irnc,parametrizeConic=>true) o8 = | -t_0^2t_1^4 | | t_0^5t_1 | | -t_1^6 | | t_0^3t_1^3 | | -t_0^6 | | t_0t_1^5 | | -t_0^4t_1^2 | 7 1 o8 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1 i9 : Iconic=ideal ring p 2 o9 = ideal(t - t t ) 1 0 2 o9 : Ideal of QQ[t , t , t ] 0 1 2 i10 : parametrize(Iconic) o10 = | -t_1^2 | | -t_0t_1 | | -t_0^2 | 3 1 o10 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1

 i11 : K=QQ; i12 : R=K[v,u,z]; i13 : 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); o13 : Ideal of R i14 : parametrize(I) o14 = | -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 o14 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1 i15 : Irnc=mapToRNC(I); o15 : Ideal of QQ[x , x , x , x ] 0 1 2 3 i16 : parametrize(Irnc) o16 = | 2t_0^2t_1 | | -t_1^3 | | 2t_0^3 | | -t_0t_1^2 | 4 1 o16 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1

Specifying J:

 i17 : K=QQ; i18 : R=K[v,u,z]; i19 : I=ideal(v^8-u^3*(z+u)^5); o19 : Ideal of R i20 : 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); o20 : Ideal of R i21 : p=parametrize(I,J) o21 = | 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 o21 : Matrix |--------------| <--- |--------------| | 2 | | 2 | | t - t t | | t - t t | \ 1 0 2 / \ 1 0 2 / i22 : parametrize(I,J,parametrizeConic=>true) o22 = | t_0^3t_1^5 | | -t_0^8 | | t_0^8-t_1^8 | 3 1 o22 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1 i23 : Irnc=mapToRNC(I,J); o23 : Ideal of QQ[x , x , x , x , x , x , x ] 0 1 2 3 4 5 6 i24 : parametrize(Irnc) o24 = | t_0^2t_2 | | -t_1t_2^2 | | t_0^3 | | -t_0t_1t_2 | | t_2^3 | | -t_0^2t_1 | | t_0t_2^2 | /QQ[t , t , t ]\ /QQ[t , t , t ]\ | 0 1 2 |7 | 0 1 2 |1 o24 : Matrix |--------------| <--- |--------------| | 2 | | 2 | | t - t t | | t - t t | \ 1 0 2 / \ 1 0 2 / i25 : parametrize(Irnc,parametrizeConic=>true) o25 = | -t_0^2t_1^4 | | t_0^5t_1 | | -t_1^6 | | t_0^3t_1^3 | | -t_0^6 | | t_0t_1^5 | | -t_0^4t_1^2 | 7 1 o25 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1 i26 : Iconic=ideal ring p 2 o26 = ideal(t - t t ) 1 0 2 o26 : Ideal of QQ[t , t , t ] 0 1 2 i27 : parametrize(Iconic) o27 = | -t_1^2 | | -t_0t_1 | | -t_0^2 | 3 1 o27 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1

 i28 : K=QQ; i29 : R=K[v,u,z]; i30 : 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); o30 : Ideal of R i31 : J=ideal(u^3+v*u*z,v*u^2+v^2*z,v^2*u-u^2*z,v^3-v*u*z); o31 : Ideal of R i32 : parametrize(I,J) o32 = | -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 o32 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1 i33 : Irnc=mapToRNC(I,J); o33 : Ideal of QQ[x , x , x , x ] 0 1 2 3 i34 : parametrize(Irnc) o34 = | 2t_0^2t_1 | | -t_1^3 | | 2t_0^3 | | -t_0t_1^2 | 4 1 o34 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1