This is generally difficult, but in some cases it is equivalent to ideal matrix phi, which does not perform any computation.
i1 : y = gens(QQ[x_0..x_5]/(x_2^2-x_2*x_3+x_1*x_4-x_0*x_5)); |
i2 : phi = rationalMap {y_4^2-y_3*y_5,-y_2*y_4+y_3*y_4-y_1*y_5, -y_2*y_3+y_3^2-y_1*y_4, -y_1*y_2+y_1*y_3-y_0*y_4, y_1^2-y_0*y_3}
o2 = -- rational map --
source: subvariety of Proj(QQ[x , x , x , x , x , x ]) defined by
0 1 2 3 4 5
{
2
x - x x + x x - x x
2 2 3 1 4 0 5
}
target: Proj(QQ[y , y , y , y , y ])
0 1 2 3 4
defining forms: {
2
x - x x ,
4 3 5
- x x + x x - x x ,
2 4 3 4 1 5
2
- x x + x - x x ,
2 3 3 1 4
- x x + x x - x x ,
1 2 1 3 0 4
2
x - x x
1 0 3
}
o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
|
i3 : time ideal phi
-- used 0.00541863 seconds
2 2
o3 = ideal (x - x x , x x - x x + x x , x x - x + x x , x x - x x +
4 3 5 2 4 3 4 1 5 2 3 3 1 4 1 2 1 3
------------------------------------------------------------------------
2
x x , x - x x )
0 4 1 0 3
QQ[x ..x ]
0 5
o3 : Ideal of -----------------------
2
x - x x + x x - x x
2 2 3 1 4 0 5
|
i4 : assert(ideal phi == ideal matrix phi) |
i5 : phi' = last graph phi
o5 = -- rational map --
source: subvariety of Proj(QQ[x , x , x , x , x , x ]) x Proj(QQ[y , y , y , y , y ]) defined by
0 1 2 3 4 5 0 1 2 3 4
{
x y - x y + x y ,
1 2 3 3 4 4
x y - x y - x y + x y ,
0 2 1 3 2 4 3 4
x y - x y + x y - x y ,
2 1 3 1 4 2 5 3
x y - x y + x y ,
1 1 2 2 5 4
x y - x y + x y ,
0 1 2 3 4 4
x y - x y + x y - x y ,
2 0 3 0 4 1 5 2
x y - x y + x y ,
1 0 3 1 4 2
x y - x y + x y ,
0 0 2 2 4 3
2
x - x x + x x - x x
2 2 3 1 4 0 5
}
target: Proj(QQ[y , y , y , y , y ])
0 1 2 3 4
defining forms: {
y ,
0
y ,
1
y ,
2
y ,
3
y
4
}
o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
|
i6 : time ideal phi'
-- used 0.245917 seconds
o6 = ideal 1
QQ[x ..x , y ..y ]
0 5 0 4
o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
(x y - x y + x y , x y - x y - x y + x y , x y - x y + x y - x y , x y - x y + x y , x y - x y + x y , x y - x y + x y - x y , x y - x y + x y , x y - x y + x y , x - x x + x x - x x )
1 2 3 3 4 4 0 2 1 3 2 4 3 4 2 1 3 1 4 2 5 3 1 1 2 2 5 4 0 1 2 3 4 4 2 0 3 0 4 1 5 2 1 0 3 1 4 2 0 0 2 2 4 3 2 2 3 1 4 0 5
|
i7 : assert(ideal phi' != ideal matrix phi') |