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

## Synopsis

• Usage:
rParametrizeRNC(I)
• Inputs:
• I, an ideal, of a rational normal curve.
• Outputs:
• a list, of a matrix phi and an ideal Ic.

## 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 :

• rParametrizeRNC(Ideal)