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Dmodules :: putWeylAlgebra(HashTable)

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

Synopsis

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])

See also