kappaAnnF1PlanarCurve -- D-annihilator of 1/f for a planar curve

Synopsis

Usage:

kappaAnnF1PlanarCurve f
Inputs:
- k, an integer, positive
- f, a ring element, a polynomial in $R = K[x_1,x_2]$
Outputs:
- I, an ideal, an ideal in the Weyl algebra $D = K<x_1,x_2,\partial_1,\partial_2>$

Description

The method uses kOrderAnnFs to efficiently compute the annihilator of $f^{-1}$, which equals the output of AnnFs after substitution $s=-1$, for a planar curve. This annihilator defines the localization: $D/I \cong R_f$. See [Castro-Jimenez, Leykin "Computing localizations iteratively" (2012)] for details.

i1 : f = reiffen(4,5)

        4    5    4
o1 = x x  + x  + x
      1 2    2    1

o1 : QQ[x ..x ]
         1   2

i2 : As = AnnFs f

              2                              2                          2   
o2 = ideal (4x dx  + 5x x dx  + 3x x dx  + 4x dx  - 16x s - 20x s, 16x x dx 
              1  1     1 2  1     1 2  2     2  2      1       2      1 2  1
     ------------------------------------------------------------------------
         3         3                     2                     2         2   
     + 4x dx  + 12x dx  - 125x x dx  - 4x dx  + 5x x dx  - 100x dx  - 64x s +
         2  1      2  2       1 2  1     1  2     1 2  2       2  2      2   
     ------------------------------------------------------------------------
                 3        4       4        3          2    2      3  2  
     500x s, 4x x dx  + 5x dx  - x dx  - 4x dx , - 64x x dx  + 36x dx  -
         2     1 2  1     2  1    2  2     1  2       1 2  1      2  1  
     ------------------------------------------------------------------------
          2            3            3  2       2  2            2      2      
     96x x dx dx  - 32x dx dx  - 36x dx  + 500x dx  + 125x x dx  - 36x dx dx 
        1 2  1  2      2  1  2      2  2       1  1       1 2  1      1  1  2
     ------------------------------------------------------------------------
                           2            2  2           2       2  2  
     + 720x x dx dx  + 100x dx dx  + 24x dx  - 29x x dx  + 260x dx  -
           1 2  1  2       2  1  2      1  2      1 2  2       2  2  
     ------------------------------------------------------------------------
                     2          2                                     2  
     368x x dx  - 72x dx  - 264x dx  - 500x dx s + 300x dx s + 1024x s  +
         1 2  1      2  1       2  2       2  1        2  2         2    
     ------------------------------------------------------------------------
                                                                  2
     2425x dx  + 125x dx  - 105x dx  + 1795x dx  + 1216x s - 8000s  - 7700s)
          1  1       2  1       1  2        2  2        2

o2 : Ideal of QQ[x ..x , dx ..dx , s]
                  1   2    1    2

i3 : A = sub(As, {last gens ring As => -1});

o3 : Ideal of QQ[x ..x , dx ..dx , s]
                  1   2    1    2

i4 : (kappa,A') = kappaAnnF1PlanarCurve f

                  2                              2                  
o4 = (2, ideal (4x dx  + 5x x dx  + 3x x dx  + 4x dx  + 16x  + 20x ,
                  1  1     1 2  1     1 2  2     2  2      1      2 
     ------------------------------------------------------------------------
          2        3         3                     2                     2   
     16x x dx  + 4x dx  + 12x dx  - 125x x dx  - 4x dx  + 5x x dx  - 100x dx 
        1 2  1     2  1      2  2       1 2  1     1  2     1 2  2       2  2
     ------------------------------------------------------------------------
          2             3  2      3                  2                 
     + 64x  - 500x , 16x dx  - 16x dx dx  + 125x x dx  - 35x x dx dx  +
          2       2     2  1      2  1  2       1 2  1      1 2  1  2  
     ------------------------------------------------------------------------
         2            2  2          2      2  2                   2     
     100x dx dx  + 12x dx  - 2x x dx  - 24x dx  + 112x x dx  - 36x dx  +
         2  1  2      1  2     1 2  2      2  2       1 2  1      2  1  
     ------------------------------------------------------------------------
        2                                                              
     84x dx  - 930x dx  + 625x dx  + 26x dx  - 893x dx  + 448x  - 3720,
        2  2       1  1       2  1      1  2       2  2       2        
     ------------------------------------------------------------------------
         4          4          3          3         2             2     
     256x dx  - 256x dx  - 500x dx  - 256x dx  + 64x x dx  - 80x x dx  +
         2  1       2  2       2  1       1  2      1 2  2      1 2  2  
     ------------------------------------------------------------------------
         3           3                      2                         2     
     100x dx  - 1024x  + 15625x x dx  + 500x dx  - 625x x dx  + 12500x dx  +
         2  2        2         1 2  1       1  2       1 2  2         2  2  
     ------------------------------------------------------------------------
                 4       5       4          3     4
     62500x , x x dx  + x dx  + x dx  + 4x x  + 5x ))
           2   1 2  2    2  2    1  2     1 2     2

o4 : Sequence

i5 : A == sub(A', ring A)

o5 = true

Ways to use kappaAnnF1PlanarCurve :

kappaAnnF1PlanarCurve(RingElement)

For the programmer

The object kappaAnnF1PlanarCurve is a method function.

kappaAnnF1PlanarCurve -- D-annihilator of 1/f for a planar curve

Synopsis

Description

See also

Ways to use kappaAnnF1PlanarCurve :

For the programmer