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80 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSMuPAD Source Code 4.3. As in 4.1, the procedure expects a list of lists encoding theset of minimal generators of a stable ideal I (ordered with respect to the lexicographicorder) and the index n of the last variable of the polynomial ring R. It outputs the Hilbertpolynomial of R/I.Input for the procedure compute Hilbert polynomial.◦ M — a list of lists encoding the set of generators of some stable ideal in lexicographicorder◦ n — the index of the last variable of the polynomial ring ROutput of the procedure compute Hilbert polynomial.◦ p — the Hilbert polynomial of R/I, where I is the ideal generated by the monomialsin Mcompute_Hilbert_polynomial:= proc(M, n)local H, g, d, p, j;begin/* First use ’compute_Hilbert_series’ with input M and nto determine the numerator of the reduced Hilbertseries and the value of d, i.e. the power of (1-t) inthe denominator of the Hilbert series */H:= compute_Hilbert_series(M, n);d:= n + 1 - (degree(op(H,1)) - degree(op(H,2)));g:= op(H,2);/* Use g and d to compute the Hilbert polynomial bythe above formula */p:= poly(0, [z]);for j from 0 to degree(g) dop:= p + poly(expand(coeff(g,j)*binomial(z+d-1-j,d-1)),[z]);end_for;return(p);end_proc:The above procedure together with 4.1 gives a possible implementation of Algorithm 2.29,where the ideas used here were stated in pseudo code.We will now compute the Hilbert polynomials of R/I 1 , R/I 2 , R/I 3 and R/I 4 for the idealsof Example 4.2.