1.2.11 `sm1.syz'
----------------

sm1.syz([F,V,W]|proc=P)
     ::  computes the syzygy of F in the ring of differential operators
     with the variable V.

RETURN
     List

P
     Number

F, V, W
     List

   * The return values is of the form [S,[G, M, T]].  Here S is the
     syzygy of F in the ring of differential operators with the
     variable V.  G is a Groebner basis of F with the weight vector W,
     and M is a matrix that translates the input matrix F to the
     Gr\"obner basis G.  T is the syzygy of the Gr\"obner basis G.  In
     summary, G = M F and S F = 0 hold as matrices.

   *    The weight vectors are given by W, which can be omitted.
     If W is not given,     the graded reverse lexicographic order will
     be used to compute Grobner basis.

   *    When a non-term order is given, the Grobner basis is computed in
       the homogenized Weyl algebra  (See Section 1.2 of the book of
     SST).     The homogenization variable h is automatically added.

     [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
     [[[y*x*dy*dx-2,-x*dx-y*dy+1]],    generators of the syzygy
      [[[x*dx+y*dy-1],[y^2*dy^2+2]],   grobner basis
       [[1,0],[y*dy,-1]],              transformation matrix
      [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]

     [294]sm1.syz([[x^2*dx^2+x*dx+y^2*dy^2+y*dy-4,x*y*dx*dy-1],[x,y],[[dx,-1,x,1]]]);
     [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
      [[[x^2*dx^2+h^2*x*dx+y^2*dy^2+h^2*y*dy-4*h^4],[y*x*dy*dx-h^4], GB
       [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
      [[1,0],[0,1],[y*dy,-x*dx]],     transformation matrix
      [[y*x*dy*dx-h^4,-x^2*dx^2-h^2*x*dx-y^2*dy^2-h^2*y*dy+4*h^4]]]]

