Singular Manual: ncHilb

| ##### 7.5.13.0. ncHilb Procedure from library `nchilbert.lib` (see nchilbert_lib). Usage: ncHilb(M,j); M a module, j an int Return: bigintvec Assume: - M is given via a Groebner basis; j is 1 or 2; - the weights of all the ring variables are positive Note: - computes the first (if j=1) or second (j=2) Hilbert series of I as bigintvec - the procedure works analogously to the commutative procedure `hilb` - If the returned vector has the form `v=(v_0,v_1,...,v_d,0)`, then the Hilbert series is `v_0 + v_1*t + ... + v_d*t^d` Example:
    LIB "nchilbert.lib";
    def A = makeUsl2(); setring A;
    ideal I = e,h-1; I = std(I);
    ncHilb(I,1); // first Hilbert series of A/I
    ==> Warning: the input generators are not a Groebner basis
    ==> The result might have no meaning
    ==> 1,-2,1,0
    ncHilb(I,2); // second Hilbert series of A/I
    ==> Warning: the input generators are not a Groebner basis
    ==> The result might have no meaning
    ==> 1,0
    ideal J = I, f^2; J = std(J);
    ncHilb(J,2);
    ==> Warning: the input generators are not a Groebner basis
    ==> The result might have no meaning
    ==> 1,1,0
    // now with weights 1,2,3
    ring r = 0,(e,f,h),wp(1,2,3);
    matrix D[3][3]; D[1,2]=-h; D[1,3]=2*e;D[2,3]=-2*f;
    def R = nc_algebra(1,D); setring R;
    ideal I = imap(A,I); I = std(I);
    ncHilb(I,1); // first weighted Hilbert series of R/I
    ==> Warning: the input generators are not a Groebner basis
    ==> The result might have no meaning
    ==> 1,-1,0,-1,1,0
    ncHilb(I,2); // second weighted Hilbert series of R/I
    ==> Warning: the input generators are not a Groebner basis
    ==> The result might have no meaning
    ==> 1,1,1,0
    matrix M[2][5] =
    e,h-1,f^2, 0,0,
    0,0,0, e,h+1;
    module G = std(M);
    print(G);
    ==> e,0,h-1,0,  f2,
    ==> 0,e,0,  h+1,0
    ncHilb(G,1); // first weighted Hilbert series of R^2/G
    ==> 2,-2,0,-2,1,1,0,1,-1,0
    ncHilb(G,2); // second weighted Hilbert series of R^2/G
    ==> 2,2,2,0,-1,-1,-1,0
See also: hilb.

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