University of Paderborn Department of Mathematics Diploma Thesis ...

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90 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSIndeed, the expansion of x 0 x 2 in I 1 just replaces the monomial generator x 0 x 2 by x 0 x 2 2,which provides the ideal (x 2 0, x 0 x 1 , x 0 x 2 2, x 3 1).(ii) The expansion of x 0 in I 2 can be computed as follows:MuPAD>> compute_expansion([[1,0,0,0,0],[0,5,0,0,0],[0,4,3,0,0],[0,4,2,6,0]],>> [1,0,0,0,0], 4);Output[[2, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0],[0, 5, 0, 0, 0], [0, 4, 3, 0, 0], [0, 4, 2, 6, 0]]As expected, the ideal obtained is (x 2 0, x 0 x 1 , x 0 x 2 , x 0 x 3 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 6 3).(iii) The expansion of x 4 1x 2 2x 6 3 in I 2 should increase the exponent of x 3 by 1:MuPAD>> compute_expansion([[1,0,0,0,0],[0,5,0,0,0],[0,4,3,0,0],[0,4,2,6,0]],>> [0,4,2,6,0], 4);Output[[1, 0, 0, 0, 0], [0, 5, 0, 0, 0], [0, 4, 3, 0, 0], [0, 4, 2, 7, 0]]As we see, the monomial x 4 1x 2 2x 6 3 has been replaced by x 4 1x 2 2x 7 3, such that we obtain the theideal (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 7 3).Similar to 4.9, we will now give the source code to compute the contraction of a monomialin a stable ideal. As in 4.9, the code is simply based on the definition of a contraction.Again, we will have to check if the set of minimal generators of a given stable ideal is stilla set of minimal generators after the contraction of some monomial. If this is not the case,we have to remove those monomials, which are divisible by other monomials in the set ofgenerators.MuPAD Source Code 4.11. As in 4.9, let M denote the set of minimal generators ofsome stable ideal ordered in descending lexicographic order, m the monomial to be contractedand n the index of the last variable in R. The procedure below expects M, m,and n as input and returns a list of lists encoding the new set of minimal generators inlexicographic order.Input for the procedure compute contraction.◦ M — a list of lists encoding the set of generators of some stable ideal in lexicographicorder
4.4. COMPUTING EXPANSIONS AND CONTRACTIONS 91◦ m — a list encoding the monomial to be contracted◦ n — the index of the last variable of the polynomial ring ROutput of the procedure compute contraction.◦ M — a list of lists, representing the set of minimal generators of the input ideal inlexicographic order after the contraction of the monomial mcompute_contraction:= proc(M, m, n)local last, mon, Ind, del, v, i, ind, j;begin/* Determine the index ’last’ of the last variableoccurring in m */for i from n+1 downto 0 doif m[i] 0 thenlast:= i;break;end_if;end_for;mon:= m;mon[last]:= mon[last] + 1;M[contains(M, mon)]:= m;for i from last+1 to n domon:= m;mon[i]:= mon[i] + 1;v:= contains(M, mon);if v > 0 thenM[v]:= null();end_if;end_for;/* Testing for minimality, i.e. removing all monomials,which are no longer minimal generators of the ideal */v:= contains(M, m);Ind:= {};del:= {};for j from 1 to last-1 dofor i from 1 to n+1 do
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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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6described in Chapter 4.Additionall
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28 CHAPTER 2. STABLE IDEALSby Lemma
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University of Paderborn Department of Mathematics Diploma Thesis ...

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