This is the basic construction for a rational map. The method is quite similar to toMap, except that it returns a RationalMap instead of a RingMap.
i1 : R := QQ[t_0..t_8]
o1 = QQ[t ..t ]
0 8
o1 : PolynomialRing
|
i2 : F = matrix{{t_0*t_3*t_5, t_1*t_3*t_6, t_2*t_4*t_7, t_2*t_4*t_8}}
o2 = | t_0t_3t_5 t_1t_3t_6 t_2t_4t_7 t_2t_4t_8 |
1 4
o2 : Matrix (QQ[t ..t ]) <--- (QQ[t ..t ])
0 8 0 8
|
i3 : phi = toMap F
o3 = map (QQ[t ..t ], QQ[x ..x ], {t t t , t t t , t t t , t t t })
0 8 0 3 0 3 5 1 3 6 2 4 7 2 4 8
o3 : RingMap QQ[t ..t ] <--- QQ[x ..x ]
0 8 0 3
|
i4 : rationalMap phi
o4 = -- rational map --
source: Proj(QQ[t , t , t , t , t , t , t , t , t ])
0 1 2 3 4 5 6 7 8
target: Proj(QQ[x , x , x , x ])
0 1 2 3
defining forms: {
t t t ,
0 3 5
t t t ,
1 3 6
t t t ,
2 4 7
t t t
2 4 8
}
o4 : RationalMap (cubic rational map from PP^8 to PP^3)
|
i5 : rationalMap F
o5 = -- rational map --
source: Proj(QQ[t , t , t , t , t , t , t , t , t ])
0 1 2 3 4 5 6 7 8
target: Proj(QQ[x , x , x , x ])
0 1 2 3
defining forms: {
t t t ,
0 3 5
t t t ,
1 3 6
t t t ,
2 4 7
t t t
2 4 8
}
o5 : RationalMap (cubic rational map from PP^8 to PP^3)
|
Multigraded rings are also permitted but in this case the method returns an object of the class MultihomogeneousRationalMap, which can be considered as an extension of the class RationalMap.
i6 : R' := newRing(R,Degrees=>{3:{1,0,0},2:{0,1,0},4:{0,0,1}})
o6 = QQ[t ..t ]
0 8
o6 : PolynomialRing
|
i7 : F' = sub(F,R')
o7 = | t_0t_3t_5 t_1t_3t_6 t_2t_4t_7 t_2t_4t_8 |
1 4
o7 : Matrix (QQ[t ..t ]) <--- (QQ[t ..t ])
0 8 0 8
|
i8 : phi' = toMap F'
o8 = map (QQ[t ..t ], QQ[x ..x ], {t t t , t t t , t t t , t t t })
0 8 0 3 0 3 5 1 3 6 2 4 7 2 4 8
o8 : RingMap QQ[t ..t ] <--- QQ[x ..x ]
0 8 0 3
|
i9 : rationalMap phi'
o9 = -- rational map --
source: Proj(QQ[t , t , t ]) x Proj(QQ[t , t ]) x Proj(QQ[t , t , t , t ])
0 1 2 3 4 5 6 7 8
target: Proj(QQ[x , x , x , x ])
0 1 2 3
defining forms: {
t t t ,
0 3 5
t t t ,
1 3 6
t t t ,
2 4 7
t t t
2 4 8
}
o9 : MultihomogeneousRationalMap (rational map from PP^2 x PP^1 x PP^3 to PP^3)
|
i10 : rationalMap F'
o10 = -- rational map --
source: Proj(QQ[t , t , t ]) x Proj(QQ[t , t ]) x Proj(QQ[t , t , t , t ])
0 1 2 3 4 5 6 7 8
target: Proj(QQ[x , x , x , x ])
0 1 2 3
defining forms: {
t t t ,
0 3 5
t t t ,
1 3 6
t t t ,
2 4 7
t t t
2 4 8
}
o10 : MultihomogeneousRationalMap (rational map from PP^2 x PP^1 x PP^3 to PP^3)
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The object rationalMap is a method function with options.