# putWeylAlgebra(HashTable) -- transforms output of diffOps into elements of Weyl algebra

## Synopsis

• Function: putWeylAlgebra
• Usage:
putWeylAlgebra m
• Inputs:
• , the output of diffOps
• Outputs:
• the differential operators as elements of the Weyl algebra

## Description

If I is an ideal of the polynomial ring R and m is the output of diffOps(I, k) then this routine returns elements of the Weyl algebra W corresponding to R whose images in W/IW are an R/I-generating set for the differential operators of order at most k.
 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal(x^2-y*z) 2 o2 = ideal(x - y*z) o2 : Ideal of R i3 : m = diffOps(I, 3) o3 = HashTable{BasisElts => | dx^3 dx^2dy dx^2dz dxdy^2 dxdydz dxdz^2 dy^3 dy^2dz dydz^2 dz^3 dx^2 dxdy dxdz dy^2 dydz dz^2 dx dy dz |} PolyGens => {-3} | 0 0 0 0 0 -2xz 0 2xy yz y2 2xz 0 z2 -yz 0 0 2x2z-2yz2 | {-3} | 0 0 0 0 0 -6yz 0 0 2xy 0 8yz y2 6xz 0 0 0 0 | {-3} | 0 0 0 0 0 -6z2 0 12yz 4xz 6xy 4z2 -yz 0 -6xz 0 0 0 | {-3} | 0 0 0 0 0 0 0 0 0 0 8xy 0 12yz 4y2 0 0 0 | {-3} | 0 0 0 0 0 -24xz 0 0 8yz 0 16xz 4xy 0 -8yz 0 0 24x2z-24yz2 | {-3} | 0 0 0 0 0 0 0 24xz 4z2 12yz 0 -4xz 0 -8z2 0 0 0 | {-3} | 0 0 0 0 0 8y2 0 0 0 0 0 0 8xy 0 0 0 0 | {-3} | 0 0 0 0 0 -24yz 0 0 0 0 16yz 0 0 8xy 0 0 0 | {-3} | 0 0 0 0 0 0 0 0 8xz 0 0 4yz 0 -16xz 0 0 0 | {-3} | 0 0 0 0 0 0 0 16z2 0 8xz 0 -4z2 0 0 0 0 0 | {-2} | 0 z y x 0 -3z 0 -3y x 0 z y 0 -x xy xz 0 | {-2} | 0 4x 0 2y 0 0 0 0 -2y 0 0 0 6z 8y 0 4yz 0 | {-2} | 0 0 4x 2z 0 0 0 0 4z 6y 0 0 0 -10z 4yz 0 0 | {-2} | 0 4y 0 0 0 24y y2 0 0 0 -12y 0 12x 0 0 4xy 0 | {-2} | 0 0 0 4x 0 -12z -2yz 0 0 0 8z 2y 0 0 0 0 0 | {-2} | 0 0 4z 0 0 0 z2 12z 0 12x 0 -6z 0 0 4xz 0 0 | {-1} | 0 0 0 1 yz 0 0 0 -1 0 0 0 0 1 -y -z -3z | {-1} | y 2 0 0 0 6 0 0 0 0 -6 0 0 0 0 0 -6x | {-1} | -z 0 2 0 2xz 0 2z -6 0 0 0 0 0 0 0 0 0 | o3 : HashTable i4 : putWeylAlgebra m o4 = | ydy-zdz zdx^2+4xdxdy+4ydy^2+2dy ydx^2+4xdxdz+4zdz^2+2dz ------------------------------------------------------------------------ xdx^2+2ydxdy+2zdxdz+4xdydz+dx yzdx+2xzdz ------------------------------------------------------------------------ -2xzdx^3-6yzdx^2dy+8y2dy^3-6z2dx^2dz-24xzdxdydz-24yzdy^2dz-3zdx^2+24ydy^ ------------------------------------------------------------------------ 2-12zdydz+6dy y2dy^2-2yzdydz+z2dz^2+2zdz ------------------------------------------------------------------------ 2xydx^3+12yzdx^2dz+24xzdxdz^2+16z2dz^3-3ydx^2+12zdz^2-6dz ------------------------------------------------------------------------ yzdx^3+2xydx^2dy+4xzdx^2dz+8yzdxdydz+4z2dxdz^2+8xzdydz^2+xdx^2-2ydxdy+ ------------------------------------------------------------------------ 4zdxdz-dx y2dx^3+6xydx^2dz+12yzdxdz^2+8xzdz^3+6ydxdz+12xdz^2 ------------------------------------------------------------------------ 2xzdx^3+8yzdx^2dy+8xydxdy^2+4z2dx^2dz+16xzdxdydz+16yzdy^2dz+zdx^2-12ydy^ ------------------------------------------------------------------------ 2+8zdydz-6dy y2dx^2dy-yzdx^2dz+4xydxdydz-4xzdxdz^2+4yzdydz^2-4z2dz^3+ydx ------------------------------------------------------------------------ ^2+2ydydz-6zdz^2 z2dx^3+6xzdx^2dy+12yzdxdy^2+8xydy^3+6zdxdy+12xdy^2 ------------------------------------------------------------------------ -yzdx^3+4y2dxdy^2-6xzdx^2dz-8yzdxdydz+8xydy^2dz-8z2dxdz^2-16xzdydz^2-xdx ------------------------------------------------------------------------ ^2+8ydxdy-10zdxdz+dx xydx^2+4yzdxdz+4xzdz^2-ydx ------------------------------------------------------------------------ xzdx^2+4yzdxdy+4xydy^2-zdx ------------------------------------------------------------------------ 2x2zdx^3-2yz2dx^3+24x2zdxdydz-24yz2dxdydz-3zdx-6xdy | 1 17 o4 : Matrix (QQ[x..z, dx, dy, dz]) <--- (QQ[x..z, dx, dy, dz])