# apolar -- the apolar map

## Synopsis

• Usage:
apolar(n,s,K)
apolar(n,s)
• Inputs:
• Optional inputs:
• Variable => ..., default value "t", specify a name for a variable
• Outputs:
• , 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