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7.7.4.0. 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:
See also:
deRhamCohom.
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