Singular Manual: deRhamCohom

| ##### 7.5.5.0. deRhamCohom Procedure from library `dmodapp.lib` (see dmodapp_lib). Usage: deRhamCohom(f[,w,eng,m]); f poly, w optional intvec, eng and m optional ints Return: ring (a Weyl Algebra) containing a list 'DR' of ideal and int Assume: Basering is commutative and over a field of characteristic 0. Purpose: computes a basis of the n-th de Rham cohomology group of the complement of the hypersurface defined by f, where n denotes the number of variables of the basering Note: The output ring is the n-th Weyl algebra. It contains a list 'DR' with two entries (ideal J and int m) such that {f^m*J[i] : i=1..size(I)} is a basis of the n-th de Rham cohomology group of the complement of the hypersurface defined by f. 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. If m is given, it is assumed to be less than or equal to the minimal integer root of the Bernstein-Sato polynomial of f. This assumption is not checked. If not specified, m is set to the minimal integer root of the Bernstein-Sato polynomial of f. 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;
    def A = deRhamCohom(f); // we see that the K-dim is 2
    setring A;
    DR;
    ==> [1]:
    ==>    _[1]=-x^3*Dx*Dy*Dz
    ==>    _[2]=-x*y*z*Dx*Dy*Dz
    ==> [2]:
    ==>    -2
See also: deRhamCohomIdeal.

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