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 |
The object kappaAnnF1PlanarCurve is a method function.