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

## Synopsis

• Usage:
kappaAnnF1PlanarCurve f
• Inputs:
• k, an integer, positive
• f, , 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