recover -- recover the binary form from its apolar ideal

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:
- a ring element, 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  
o1 = 810t  + 405t t  + 135t t  + 315t t  + 140t t  + 126t t  + 126t t  +
         0       0 1       0 1       0 1       0 1       0 1       0 1  
     ------------------------------------------------------------------------
         7
     420t
         1

o1 : QQ[t ..t ]
         0   1

i2 : I = apolar F

                      4              3                2 2                3  
o2 = ideal (201527032t  - 3753654030t t  - 1695069684t t  + 7318635282t t  -
                      0              0 1              0 1              0 1  
     ------------------------------------------------------------------------
               4        3 2       2 3           4        5
     258009867t , 47922t t  - 396t t  - 80634t t  + 3005t )
               1        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 (47922t t  - 396t t  - 80634t t  + 3005t , 15974t t  - 136818t t 
                  0 1       0 1         0 1        1        0 1          0 1
     ------------------------------------------------------------------------
               4        5       5          2 3            4        5
     + 79479t t  - 1794t , 2282t  - 281340t t  + 176265t t  - 3570t )
             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  
o4 = 810t  + 405t t  + 135t t  + 315t t  + 140t t  + 126t t  + 126t t  +
         0       0 1       0 1       0 1       0 1       0 1       0 1  
     ------------------------------------------------------------------------
         7
     420t
         1

o4 : QQ[t ..t ]
         0   1

i5 : recover I5

         7       6         5 2       4 3       3 4       2 5         6  
o5 = 810t  + 405t t  + 135t t  + 315t t  + 140t t  + 126t t  + 126t t  +
         0       0 1       0 1       0 1       0 1       0 1       0 1  
     ------------------------------------------------------------------------
         7
     420t
         1

o5 : QQ[t ..t ]
         0   1

Ways to use recover:

recover(Ideal)
recover(RingElement,RingElement)

For the programmer

The object recover is a method function.

The source of this document is in CoincidentRootLoci/documentation.m2:219:0.

recover -- recover the binary form from its apolar ideal

Description

See also

Ways to use recover:

For the programmer