##### 7.5.5.0. deRhamCohomIdeal

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

Usage:
  deRhamCohomIdeal (I[,w,eng,k,G]);
  I ideal, w optional intvec, eng and k optional ints, G optional ideal

Return:
  ideal

Assume:
  The basering is the n-th Weyl algebra D over a field of characteristic
  zero 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).
  Further, assume that I is of special kind, namely let f in K[x] and
  consider the module K[x,1/f]f^m, where m is smaller than or equal to
  the minimal integer root of the Bernstein-Sato polynomial of f.
  Since this module is known to be a holonomic D-module, it has a cyclic
  presentation D/I.

Purpose:
  computes a basis of the n-th de Rham cohomology group of the complement
  of the hypersurface defined by f

Note:
  The elements of the basis are of the form f^m*p, where p runs over the
  entries of the returned ideal.
  If I does not satisfy the assumptions described above, the result might
  have no meaning. Note that I can be computed with `annfs`.
  If w is an intvec with exactly n strictly positive entries, w is used
  in the computation. Otherwise, and by default, w is set to (1,...,1).
  If eng<>0, `std` is used for Groebner basis computations,
  otherwise, and by default, `slimgb` is used.
  Let F(I) denote the Fourier transform of I wrt w.
  An integer smaller than or equal to the minimal integer root of the
  b-function of F(I) wrt the weight (-w,w) can be specified via the
  optional argument k.
  The optional argument G is used for specifying a Groebner Basis of F(I)
  wrt the weight (-w,w), that is, the initial form of G generates the
  initial ideal of F(I) wrt the weight (-w,w).
  Further note, that the assumptions on I, k and G (if given) are not
  checked.

Theory:
  (SST) pp. 232-235

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,z),dp;
    poly F = x^3+y^3+z^3;
    bfctAnn(F);            // Bernstein-Sato poly of F has minimal integer root -2
    ==> [1]:
    ==>    _[1]=-1
    ==>    _[2]=-4/3
    ==>    _[3]=-5/3
    ==>    _[4]=-2
    ==> [2]:
    ==>    2,1,1,1
    def W = annRat(1,F^2); // so we compute the annihilator of 1/F^2
    setring W; W;          // Weyl algebra, contains LD = Ann(1/F^2)
    ==> // coefficients: QQ considered as a field
    ==> // number of vars : 6
    ==> //        block   1 : ordering dp
    ==> //                  : names    x y z Dx Dy Dz
    ==> //        block   2 : ordering C
    ==> // noncommutative relations:
    ==> //    Dxx=x*Dx+1
    ==> //    Dyy=y*Dy+1
    ==> //    Dzz=z*Dz+1
    LD;                    // K[x,y,z,1/F]F^(-2) is isomorphic to W/LD as W-module
    ==> LD[1]=x*Dx+y*Dy+z*Dz+6
    ==> LD[2]=z^2*Dy-y^2*Dz
    ==> LD[3]=z^2*Dx-x^2*Dz
    ==> LD[4]=y^2*Dx-x^2*Dy
    ==> LD[5]=x^3*Dz+y^3*Dz+z^3*Dz+6*z^2
    ==> LD[6]=x^3*Dy+y^3*Dy+y^2*z*Dz+6*y^2
    deRhamCohomIdeal(LD);  // we see that the K-dim is 2
    ==> _[1]=-x^3*Dx*Dy*Dz
    ==> _[2]=-x*y*z*Dx*Dy*Dz

See also: deRhamCohom.

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