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# recover -- recover the binary form from its apolar ideal

## Synopsis

• Usage:
recover I
• Inputs:
• I, an ideal, the apolar ideal of a binary form $F\in K[x,y]$, or one of its homogeneous components of sufficiently large degree
• Outputs:
• , the binary form $F$ (up to a multiplicative constant)

## Description

 i1 : F = randomBinaryForm 7 7 6 5 2 4 3 3 4 2 5 6 7 o1 = 18t + 2t t + 9t t + 2t t + 4t t + 3t t + 6t t + 3t 0 0 1 0 1 0 1 0 1 0 1 0 1 1 o1 : QQ[t ..t ] 0 1 i2 : I = apolar F 4 3 2 2 3 4 o2 = ideal (75621t + 740842t t - 3587636t t - 2099787t t + 741116t , 0 0 1 0 1 0 1 1 ------------------------------------------------------------------------ 3 2 2 3 4 5 1355t t - 5785t t - 3840t t + 1321t ) 0 1 0 1 0 1 1 o2 : Ideal of QQ[t ..t ] 0 1 i3 : I5 = apolar(F,5) 3 2 2 3 4 5 4 2 3 o3 = ideal (1355t t - 5785t t - 3840t t + 1321t , 1355t t - 7610t t - 0 1 0 1 0 1 1 0 1 0 1 ------------------------------------------------------------------------ 4 5 5 2 3 4 5 5t t + 338t , 271t - 47505t t - 33770t t + 11872t ) 0 1 1 0 0 1 0 1 1 o3 : Ideal of QQ[t ..t ] 0 1 i4 : recover I 7 6 5 2 4 3 3 4 2 5 6 7 o4 = 18t + 2t t + 9t t + 2t t + 4t t + 3t t + 6t t + 3t 0 0 1 0 1 0 1 0 1 0 1 0 1 1 o4 : QQ[t ..t ] 0 1 i5 : recover I5 7 6 5 2 4 3 3 4 2 5 6 7 o5 = 18t + 2t t + 9t t + 2t t + 4t t + 3t t + 6t t + 3t 0 0 1 0 1 0 1 0 1 0 1 0 1 1 o5 : QQ[t ..t ] 0 1