##### 7.5.5.0. GBWeight

Procedure from library `dmodapp.lib` (see dmodapp_lib).

Usage:
  GBWeight(I,u,v [,s,t,w]);
  I ideal, u,v intvecs, s,t optional ints, w an optional intvec

Return:
  ideal, Groebner basis of I w.r.t. the weights u and v

Assume:
  The basering is the n-th Weyl algebra over a field of characteristic 0
  and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1
  holds, i.e. the sequence of variables is given by
  x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator
  belonging to x(i).

Purpose:
  computes a Groebner basis with respect to given weights

Note:
  The weights u and v are understood as weight vectors for x(i) and D(i),
  respectively. According to (SST), one computes the homogenization of a
  given ideal relative to (u,v), then one computes a Groebner basis and
  returns the dehomogenization of the result.
  If s<>0, `std` is used for Groebner basis computations,
  otherwise, and by default, `slimgb` is used.
  If t<>0, a matrix ordering is used for Groebner basis computations,
  otherwise, and by default, a block ordering is used.
  If w is given and consists of exactly 2*n strictly positive entries,
  w is used for constructing the weighted homogenized Weyl algebra,
  see Noro (2002). Otherwise, and by default, the homogenization weight
  (1,...,1) is used.

Display:
  If printlevel=1, progress debug messages will be printed,
  if printlevel>=2, all the debug messages will be printed.

Example:

    LIB "dmodapp.lib";
    ring r = 0,(x,y,Dx,Dy),dp;
    def D2 = Weyl();
    setring D2;
    ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6;
    intvec u = -2,-3;
    intvec v = -u;
    GBWeight(I,u,v);
    ==> _[1]=2*x*Dx+3*y*Dy+6
    ==> _[2]=3*x^2*Dy+2*y*Dx
    ==> _[3]=9*x*y*Dy^2-4*y*Dx^2+15*x*Dy
    ==> _[4]=27*y^2*Dy^3+8*y*Dx^3+135*y*Dy^2+105*Dy
    ideal J = std(I);
    GBWeight(J,u,v); // same as above
    ==> _[1]=2*x*Dx+3*y*Dy+6
    ==> _[2]=3*x^2*Dy+2*y*Dx
    ==> _[3]=9*x*y*Dy^2-4*y*Dx^2+15*x*Dy
    ==> _[4]=27*y^2*Dy^3+8*y*Dx^3+135*y*Dy^2+105*Dy
    u = 0,1;
    GBWeight(I,u,v);
    ==> _[1]=2*x*Dx+3*y*Dy+6
    ==> _[2]=3*x^2*Dy+2*y*Dx
    ==> _[3]=-x^3*Dx+y^2*Dx-3*x^2

User manual for Singular version 4.4.1, 2025, generated by texi2html.