In most cases, the command rationalMap(I,d,e) yields the same output as rationalMap(saturate(I^e),d), but the former is implemented using pure linear algebra.
The command rationalMap I is basically equivalent to rationalMap(I,max degrees I).
In the following example, we calculate the rational map defined by the linear system of cubic hypersurfaces in $\mathbb{P}^6$ having double points along a Veronese surface $V\subset\mathbb{P}^5\subset\mathbb{P}^6$.
i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
ZZ
o2 : Ideal of -----[x ..x ]
33331 0 6
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i3 : time phi = rationalMap(V,3,2)
-- used 0.152036 seconds
o3 = -- rational map --
ZZ
source: Proj(-----[x , x , x , x , x , x , x ])
33331 0 1 2 3 4 5 6
ZZ
target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y , y , y , y ])
33331 0 1 2 3 4 5 6 7 8 9 10 11 12 13
defining forms: {
3
x ,
6
2
x x ,
5 6
2
x x ,
4 6
2
x x ,
3 6
2
x x ,
2 6
2
x x ,
1 6
2
x x ,
0 6
2
x x - x x x ,
4 6 3 5 6
x x x - x x x ,
2 4 6 1 5 6
x x x - x x x ,
2 3 6 1 4 6
2
x x - x x x ,
2 6 0 5 6
x x x - x x x ,
1 2 6 0 4 6
2
x x - x x x ,
1 6 0 3 6
2 2 2
x x - 2x x x + x x + x x - x x x
2 3 1 2 4 0 4 1 5 0 3 5
}
o3 : RationalMap (cubic rational map from PP^6 to PP^13)
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i4 : describe phi!
o4 = rational map defined by forms of degree 3
source variety: PP^6
target variety: PP^13
image: 6-dimensional variety of degree 16 in PP^13 cut out by 21 hypersurfaces of degree 2
dominance: false
birationality: false
degree of map: 1
projective degrees: {1, 3, 6, 12, 16, 16, 16}
number of minimal representatives: 1
dimension base locus: 4
degree base locus: 3
coefficient ring: ZZ/33331
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