University of Paderborn Department of Mathematics Diploma Thesis ...

84 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSOutput. Set of minimal generators of the ideal L p .Proof. (Correctness) By Remark 1.11 and Theorem 2.25, the algorithm is correct.MuPAD Source Code 4.7. A possible implementation of Algorithm 4.6 is the following,where we use lists to represent monomials as in 4.1 and 4.3. The procedure expects aHilbert polynomial p and the index of the last variable of R and outputs the set of generatorsof the unique lexicographic ideal L p .Input for the procedure compute L p.◦ p — the Hilbert polynomial of R/I◦ n — the index of the last variable of the polynomial ring ROutput of the procedure compute L p.◦ L p — a list of lists encoding the set of minimal generators of the lexicographic idealL p associated to pcompute_L_p:= proc(p, n)local indet, coeff, deg, m, a, L_p, monom, cnt, i;beginindet:= op(p)[2][1];deg:= degree(p);m:= [0 $ deg + 1];a:= m;/* Compute the m_i’s */for i from deg downto 1 docoeff:= lcoeff(p);m[i+1]:= i! * coeff;p:= p - poly(expand(binomial(indet + i, i+1)),[indet])+ poly(expand(binomial(indet + i - m[i+1], i+1)),[indet]);end_for;m[1]:= op(p)[1];/* Compute the a_i’s */a[deg + 1]:= m[deg + 1];for i from deg - 1 downto 0 do