University of Paderborn Department of Mathematics Diploma Thesis ...

102 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSIdeal:= compute_expansion(Ideal, mon, n);if not contains(IdealSet, Ideal) thenIdealSet:= IdealSet union {Ideal};included:= TRUE;end_if;elsenext;end_if;end_for;/* Compute all other saturated stable ideals with thesame Hilbert polynomial and the same double saturationas the input ideal J recursively. */if included = TRUE thenfor posIdeal in IdealSet dotmp:= compute_ideals(posIdeal, n);IdealSet:= IdealSet union tmp;end_for;end_if;end_if;return(IdealSet union {J});end_proc:We demonstrate the use of procedure 4.17 by an example adapted from [16], Chapter 3,Example 2, pp. 25,26 :Example 4.18. Let R := K[x 0 , x 1 , x 2 , x 3 , x 4 ] and p(z) := 4z + 1. First, we compute theunique lexicographic ideal L p via procedure 4.7:MuPAD>> L_p:= compute_L_p(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]]i.e. L p = (x 0 , x 1 , x 5 2, x 4 2x 3 3). Now we can use procedure 4.17 to compute all saturated stableideals with the same double saturation and the same Hilbert polynomial as L p .