Singular Manual: integralModule

| ##### 7.7.5.0. integralModule Procedure from library `dmodapp.lib` (see dmodapp_lib). Usage: integralModule(I,w,[,eng,m,G]); I ideal, w intvec, eng and m optional ints, G optional ideal Return: ring (a Weyl algebra) containing a module 'intMod' 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). Further, assume that I is holonomic and that w is n-dimensional with non-negative entries. Purpose: computes the integral module of a holonomic ideal w.r.t. the subspace defined by the variables corresponding to the non-zero entries of the given intvec Note: The output ring is the Weyl algebra defined by the zero entries of w. It contains a module 'intMod' being the integral module of I wrt w. If there are no zero entries, the input ring is returned. 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 w.r.t. w. The minimal integer root of the b-function of F(I) w.r.t. the weight (-w,w) can be specified via the optional argument m. 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) w.r.t. the weight (-w,w). Further note, that the assumptions on m and G (if given) are not checked. 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,b,Dx,Db),dp;
    def D2 = Weyl();
    setring D2;
    ideal I = x*Dx+2*b*Db+2, x^2*Dx+b*Dx+2*x;
    intvec w = 1,0;
    def im = integralModule(I,w);
    setring im; im;
    ==> // coefficients: QQ
    ==> // number of vars : 2
    ==> //        block   1 : ordering C
    ==> //        block   2 : ordering dp
    ==> //                  : names    b Db
    ==> // noncommutative relations:
    ==> //    Dbb=b*Db+1
    print(intMod);
    ==> 2*b*Db+1,0,
    ==> 0,       b*Db

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