University of Paderborn Department of Mathematics Diploma Thesis ...

4.7. COMPUTING ALL HILBERT FUNCTIONS TO A HILBERT POLYNOMIAL 121end_for:end_for:for i from d+2 to G_bound - 1 dovalues[i]:= coeff(H,i-1):for j from 1 to d dovalues[i]:= values[i] - (values[i-j] * coeffs[d+1-j])end_for:end_for:/* Now compute the Hilbert function with the Hilbert polynomial. */for i from G_bound to upper_bound+1 dovalues[i]:= p(i-1):end_for:return(values);end_proc:Example 4.29. We determine the values of the Hilbert functions associated to the Hilbertseries computed in Example 4.24.MuPAD>> compute_Hilbert_function(poly(-t^7+t^4+t^3+t^2+t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^4+t^3+t^2+2*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-2*t^6+t^5+t^4+t^3+t^2+t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^3+2*t^2+2*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^6+t^4+t^3+2*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^5+t^4+t^3-t^2+3*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(3*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^5+t^4+t^2+2*t+1,[t]),2,10,4);Output[1, 3, 6, 10, 15, 20, 25, 29, 33, 37, 41][1, 4, 8, 13, 17, 21, 25, 29, 33, 37, 41][1, 3, 6, 10, 15, 21, 25, 29, 33, 37, 41][1, 4, 9, 13, 17, 21, 25, 29, 33, 37, 41][1, 4, 7, 11, 16, 21, 25, 29, 33, 37, 41][1, 5, 8, 12, 17, 21, 25, 29, 33, 37, 41][1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41][1, 4, 8, 12, 17, 21, 25, 29, 33, 37, 41]