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 parametizeConic=>true is given and C has a rational point then the conic is parametrized hence pI is over ℙ1.
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
|