University of Paderborn Department of Mathematics Diploma Thesis ...

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106 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSprint(Unquoted, "<strong>of</strong> all ideals to the given Hilbert polynomial");print(Unquoted, "equals the whole ring");return(M);end_if;if deg = 1 and args(0) = 3 and args(3) = hold(All) thencnt:= nops(M);Sat:= M;end_if;/* Successively compute all saturated ideals */for i from d+1 downto 2 do/* Consider the elements <strong>of</strong> M as ideals in the polynomialring with one more variable (lifting-process) */M:= map(M, Ideal -> map(Ideal, ind -> [op(ind), 0]));/* Compute the Hilbert polynomials and perform expansionsif necessary. */p1:= pList[i-1];tmp:= {};for Ideal in M dop2:= compute_Hilbert_polynomial(Ideal, n-i+2);steps:= p1 - p2;if degree(steps) = 0 and op(steps, 1) > 0 thenM:= M minus {Ideal};J:= Ideal;for j from 1 to op(steps, 1) doJ:= compute_expansion(J, J[nops(J)], n-i+2);end_for;tmp:= tmp union {J};elif degree(steps) > 0 or op(steps, 1) < 0 thenM:= M minus {Ideal};if i = 2 and deg > 1 thenId:= map(Ideal, ind -> [ind[k] $ k = 1..n]);Sat:= Sat minus {Id};end_if;end_if;end_for;
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],
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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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8 CHAPTER 1. NOTATIONS AND PREREQUI
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