We illustrate two ways of using this method.
1. Given a binary form $F$ of degree $n$ we can obtain a basis for the space of forms of degree $s$ that annihilate $F$, say for example $(n,s)=(6,4)$.
i1 : (n,s) = (6,4) o1 = (6, 4) o1 : Sequence |
i2 : F = randomBinaryForm n
6 5 4 2 3 3 2 4 5 6
o2 = 18t + 2t t + 9t t + 2t t + 4t t + 3t t + 6t
0 0 1 0 1 0 1 0 1 0 1 1
o2 : QQ[t , t ]
0 1
|
i3 : phi = apolar(n,s)
o3 = -- rational map --
source: Proj(QQ[t , t , t , t , t , t , t ])
0 1 2 3 4 5 6
target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4
{
t t - t t + t t ,
2,3 1,4 1,3 2,4 1,2 3,4
t t - t t + t t ,
2,3 0,4 0,3 2,4 0,2 3,4
t t - t t + t t ,
1,3 0,4 0,3 1,4 0,1 3,4
t t - t t + t t ,
1,2 0,4 0,2 1,4 0,1 2,4
t t - t t + t t
1,2 0,3 0,2 1,3 0,1 2,3
}
defining forms: {
3 2 2
- 24t + 48t t t - 24t t - 24t t + 24t t t ,
4 3 4 5 2 5 3 6 2 4 6
2 2 2
16t t - 16t t - 16t t t + 16t t + 16t t t - 16t t t ,
3 4 3 5 2 4 5 1 5 2 3 6 1 4 6
2 2
- 4t t + 4t t t + 4t t t - 4t t - 4t t t + 4t t t ,
2 4 2 3 5 1 4 5 0 5 1 3 6 0 4 6
2 2 2
- 24t t + 24t t + 24t t t - 24t t t - 24t t + 24t t t ,
3 4 2 4 2 3 5 1 4 5 2 6 1 3 6
2 2
6t t t - 6t t - 6t t + 6t t t + 6t t t - 6t t t ,
2 3 4 1 4 2 5 0 4 5 1 2 6 0 3 6
2 2
- 4t t + 4t t t + 4t t t - 4t t t - 4t t + 4t t t ,
2 4 1 3 4 1 2 5 0 3 5 1 6 0 2 6
3 2 2
96t - 192t t t + 96t t + 96t t - 96t t t ,
3 2 3 4 1 4 2 5 1 3 5
2 2 2
- 24t t + 24t t + 24t t t - 24t t - 24t t t + 24t t t ,
2 3 2 4 1 3 4 0 4 1 2 5 0 3 5
2 2 2
16t t - 16t t - 16t t t + 16t t t + 16t t - 16t t t ,
2 3 1 3 1 2 4 0 3 4 1 5 0 2 5
3 2 2
- 24t + 48t t t - 24t t - 24t t + 24t t t
2 1 2 3 0 3 1 4 0 2 4
}
o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
|
i4 : P = switch plucker phi switch switch F
3 2 2 3 4 4 2 2
o4 = ideal (20458t t - 4533t t - 52230t t + 4213t , 10229t - 325779t t +
0 1 0 1 0 1 1 0 0 1
------------------------------------------------------------------------
3 4
99734t t + 5145t )
0 1 1
o4 : Ideal of QQ[t , t ]
0 1
|
i5 : diff(gens P,F) == 0 o5 = true |
2. We can recover the form $F$ from the above output.
i6 : switch phi^* plucker switch P
6 5 4 2 3 3 2 4 5 6
o6 = 18t + 2t t + 9t t + 2t t + 4t t + 3t t + 6t
0 0 1 0 1 0 1 0 1 0 1 1
o6 : QQ[t , t ]
0 1
|
i7 : oo == F o7 = true |
The object apolar is a method function with options.