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Parametrization :: rParametrizeRNC

rParametrizeRNC -- Compute a rational parametrization of a rational normal curve.

Synopsis

Description

Compute a rational parametrization of a rational normal curve C defined by I.

phi contains the rational parametrization of C over ℙ1 if the degree of C is odd or over a conic if the degree of C is even.

Ic is the 0-ideal for odd degree or the ideal of the conic for even degree.

i1 : K=QQ;
i2 : R=K[v,u,z];
i3 : I0=ideal(v^8-u^3*(z+u)^5);

o3 : Ideal of R
i4 : J=ideal matrix {{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 : I=mapToRNC(I0,J)

                                                                            
o5 = ideal (x x  - x x , x x  - x x , x x  - x x , x x  - x x , x x  - x x ,
             4 5    3 6   3 5    2 6   1 5    0 6   3 4    1 6   2 4    0 6 
     ------------------------------------------------------------------------
             2   2                              2                           
     x x  - x , x  - x x , x x  - x x , x x  - x , x x  - x x , x x  - x x ,
      0 4    6   3    0 6   2 3    0 5   1 3    6   0 3    5 6   1 2    5 6 
     ------------------------------------------------------------------------
             2   2                       2
     x x  - x , x  - x x , x x  - x x , x  - x x )
      0 2    5   1    4 6   0 1    3 6   0    2 6

o5 : Ideal of QQ[x , x , x , x , x , x , x ]
                  0   1   2   3   4   5   6
i6 : rParametrizeRNC(I)

                             2
o6 = {| t_0^2t_2   |, ideal(t  - t t )}
      | -t_1t_2^2  |         1    0 2
      | t_0^3      |
      | -t_0t_1t_2 |
      | t_2^3      |
      | -t_0^2t_1  |
      | t_0t_2^2   |

o6 : List

Caveat

Perhaps better return just phi in the quotient ring by Ic.

Ways to use rParametrizeRNC :