University of Paderborn Department of Mathematics Diploma Thesis ...

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100 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSOutput. The set M, i.e. all saturated stable ideals with the same double saturation andthe same Hilbert polynomial as I.Proof. (Correctness) By Theorem 3.32, the algorithm finds all saturated stable ideals withthe same double saturation and the same Hilbert polynomial. The set M contains – ateach step of computation – a subset of all saturated stable ideals with the same Hilbertpolynomial and the same double saturation as the input ideal I. The number of elements ofM increases strictly monotonic within each round of the algorithm. Since the number of allsaturated stable ideals with the same Hilbert polynomial and the same double saturationas the ideal I is finite, it terminates.Note that we can use 4.7 to compute the unique lexicographic ideal L p associated to theHilbert polynomial p(z). Thus, if we combine 4.7 with an implementation of the abovealgorithm, we obtain all saturated stable ideals with the same Hilbert polynomial andthe same double saturation as the unique lexicographic ideal L p associated to p(z). TheMuPAD source code is listed below. Steps 2 and 3 of the above algorithm are realized byusing the procedures presented in 4.13 and 4.14, whereas step 4 is done by recursive callsof the procedure itself.MuPAD Source Code 4.17. This procedure expects two arguments: J has to be a listof lists encoding a saturated stable ideal and n is the index of the last variable of R. Theoutput is a set of lists, each encoding the set of minimal generators of a stable ideal.Input for the procedure compute ideals.◦ J — a list of lists encoding the set of generators of some saturated stable ideal inlexicographic order◦ n — the index of the last variable of the polynomial ring ROutput of the procedure compute ideals.◦ IdealSet — a set of lists of lists, encoding all saturated stable ideals with the samedouble saturation and the same Hilbert polynomial as the input ideal Jcompute_ideals:= proc(J, n)local expandable, i, j, Ideal, mon, contractMon,posIdeal, IdealSet, tmp, included;beginIdealSet:= {};/* Find all monomial generators in J with the last variablex_{n-1}. These are the monomials, which will be usedfor the contractions. */
4.5. COMPUTING STABLE IDEALS TO A GIVEN HILBERT POLYNOMIAL 101contractMon:= 0;for j from nops(J) downto 1 doif J[j][n] > 0 thencontractMon:= J[j];break;end_if;end_for;/* If there is no monomial contractible, return theset, which contains only the input ideal */if contractMon = 0 thenreturn({J});end_if:/* Find all expandable monomials in the set of generators.Use procedure ’test_expandable’ to check, whether any ofthe monomials generators is expandable. */expandable:= {};for i from 1 to nops(J) doif test_expandable(J, J[i], n) = TRUE andcontains(J, contractMon) > i thenexpandable:= expandable union {J[i]};end_if;end_for;contractMon[n]:= contractMon[n] - 1;included:= FALSE;/* Perform all possible pairs of contractions andexpansions, if the set ’expandable’ of expandablemonomials is not the empty set*/if expandable {} thenfor mon in expandable doif test_contractible(J, contractMon, n) = TRUE andtest_expandable(compute_contraction(J, contractMon, n),mon, n) = TRUE thenIdeal:= compute_contraction(J, contractMon, n);
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University of PaderbornDepartment o
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”Im großen Garten der Geometriek
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AbstractThe main result of this the
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Glossary of MuPAD procedures 129Bib
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This thesis is organized in four ch
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