University of Paderborn Department of Mathematics Diploma Thesis ...

4.5. COMPUTING STABLE IDEALS TO A GIVEN HILBERT POLYNOMIAL 107/* Include the lately computed ideals into the set M. */M:= M union tmp;MM:= M;/* Use procedure ’compute_ideals’ to find all saturatedideals with the same Hilbert polynomial and the samedouble saturation recursively. */for Ideal in MM doM:= M union compute_ideals(Ideal, n-i+2);end_for;/* If i = 3 and deg > 1 the set M contains all possibledouble saturations. */if i = 3 and deg > 1 thenSat:= M;end_if:end_for;if args(0) = 3 and args(3) = hold(All) thenreturn(M, Sat, nops(Sat))elsereturn(M);end_if;end_proc:The example below is again adapted from [16], Chapter 3, Example 2, pp. 25,26.Example 4.21. Let p(z) = 4z + 1 and R := K[x 0 , x 1 , x 2 , x 3 , x 4 ]. Then we can computeall saturated stable ideals J ⊂ R, such that p R/J (z) = p(z), as follows:MuPAD>> compute_all_ideals(poly(4*z + 1, [z]), 4);Output{[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 5, 0, 0], [0, 0, 4, 3, 0]],[[1, 0, 0, 0, 0], [0, 2, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 4, 0, 0],[0, 0, 3, 1, 0]], [[1, 0, 0, 0, 0], [0, 2, 0, 0, 0], [0, 1, 1, 0, 0],