apolar -- the apolar map

Synopsis

Usage:

apolar(n,s,K)

apolar(n,s)
Inputs:
- n, an integer
- s, an integer
- K, a ring, the coefficient ring (optional with default value QQ)
Optional inputs:
- Variable => ..., default value "t", specify a name for a variable
Outputs:
- a rational map, the map from $\mathbb{P}^n$ to $Grass(2s-n-1,s)$ that sends a generic binary form $F\in K[x,y]$ of degree $n$ to the set of binary forms of degree $s$ that annihilate $F$, which can be identified with a $(2s-n-1)$-dimensional linear subspace of $\mathbb{P}^s$.

Description

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

Ways to use apolar:

apolar(ZZ,ZZ)
apolar(ZZ,ZZ,Ring)
apolar(RingElement) -- the apolar ideal
apolar(RingElement,ZZ) -- homogeneous components of the apolar ideal

For the programmer

The object apolar is a method function with options.

apolar -- the apolar map

Synopsis

Description

See also

Ways to use apolar:

For the programmer