University of Paderborn Department of Mathematics Diploma Thesis ...

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88 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSOutput of the procedure compute expansion.◦ M — a list of lists, representing the set of minimal generators of the input ideal inlexicographic order after the expansion of the monomial mcompute_expansion:= proc(M, m, n)local newMonomials, last, mon, Ind, del, v, i, ind, j, monomial;begin/* Compute the set of new monomials to be included intothe set M of minimal generators. Hence, first computethe index ’last’ of the last variable occurring in m */for i from n+1 downto 0 doif m[i] 0 thenlast:= i;break;end_if;end_for;newMonomials:= [0 $ n-last+1];for i from last to n domon:= m;mon[i]:= mon[i] + 1;newMonomials[i-last+1]:= mon;end_for;/* Testing for minimality, i.e. removing all monomials,which are no longer minimal generators of the ideal */Ind:= {};for monomial in newMonomials dofor j from contains(M,m) + 1 to nops(M) dodel:= {};for i from 1 to n+1 doif monomial[i] - M[j][i] > 0 thendel:= del union {1};elif monomial[i] - M[j][i] < 0 thendel:= del union {-1};end_if;end_for;if contains(del, 1) = FALSE orcontains(del, -1) = FALSE then
4.4. COMPUTING EXPANSIONS AND CONTRACTIONS 89Ind:= Ind union {j};end_if;end_for;end_for;v:= 0;for ind in Ind doM[ind-v]:= null();v:= v+1;end_for;/* Delete m from M and insert the set of newminimal generators into M */M:= [ M[j] $ j = 1..contains(M,m)-1,op(newMonomials),M[j] $ j = contains(M,m)+1..nops(M) ];return(M);end_proc:Although the above lines might look a little complicated, this is not really the case. Thecomputation of the set of minimal generators after performing the expansion of somemonomial m uses nothing more than the original definition of the expansion. The only fact,which makes the source code look more complicated than perhaps expected is that we haveto check if all monomials, which are minimal generators in the set M before the expansion,are still minimal generators after expanding m. To guarantee that the procedure alwaysreturns a set of minimal generators again, it suffices to consider all monomials m ′ ∈ M tothe right of m. Hence, we check for all of these monomials m ′ in M, if they are divisibleby any of the monomials, which are included into M by the expansion of m. If this is thecase, we can remove such elements m ′ from M. Some examples will illustrate the use ofthis new procedure.Example 4.10. We consider the ideals I 1 := (x 2 0, x 0 x 1 , x 0 x 2 , x 3 1) ⊂ K[x 0 , x 1 , x 2 , x 3 ] andI 2 := (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 6 3) ⊂ K[x 0 , x 1 , x 2 , x 3 , x 4 ].(i) First we expand the monomial x 0 x 2 in I 1 :MuPAD>> compute_expansion([[2,0,0,0],[1,1,0,0],[1,0,1,0],[0,3,0,0]],>> [1,0,1,0], 3);Output[[2, 0, 0, 0], [1, 1, 0, 0], [1, 0, 2, 0], [0, 3, 0, 0]]
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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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