University of Paderborn Department of Mathematics Diploma Thesis ...

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104 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSMuPAD>> L_p:= compute_L_p(poly(2*z^2 + 2*z + 1, [z]), 4):>> nops(compute_ideals(L_p, 4));Output77In the next section, we take advantage of procedure 4.17 to compute all Hilbert seriesassociated to a given Hilbert polynomial.The last part of this section is dedicated to an implementation of Algorithm 3.45, i.e. thenext procedure can be used to compute all saturated stable ideals to a given Hilbertpolynomial.MuPAD Source Code 4.20. Let p be the Hilbert polynomial of R/I for I ⊂ R a saturatedstable ideal and n the index of the last variable of R. The procedure below outputsall saturated stable ideals to the Hilbert polynomial p in R. If one uses the option ’All’as third argument for the procedure, it additionally returns the set and the number of alldouble saturations of the saturated stable ideals associated to the Hilbert polynomial p.Input for the procedure compute all ideals.◦ p — the Hilbert polynomial of R/I◦ n — the index of the last variable of the polynomial ring R◦ All — an optional, third input parameter (for details, see description of the outputof the procedure below)Output of the procedure compute all ideals.◦ M — a set of lists of lists, encoding all saturated stable ideals J ⊂ R with p R/J (z) =p(z).◦ Sat — a set of lists of lists, encoding all possible double saturations (as ideals in ¯R :=K[x 0 , . . . , x n−1 ]). Note that Sat will only be returned, if the third input parameterAll is used.◦ cnt — the number of elements of the set Sat. Note that cnt will only be returned,if the third input parameter All is used.compute_all_ideals:= proc(p, n /*, All */)local d, pList, i, L, j, M, MM, Ideal, p1, p2, steps, J, tmp,Sat, deg, Id, k;begin
4.5. COMPUTING STABLE IDEALS TO A GIVEN HILBERT POLYNOMIAL 105d:= 1;deg:= degree(p);pList:= [p, 0 $ deg];/* ’pList’ will contain the set of the ’Delta polynomials’, i.e.pList = [ p,Delta^1(z) = p(z) - p(z-1),Delta^2(z) = Delta^1(z) - Delta^1(z-1). . .. . .. . .Delta^m(z) = Delta^(m-1)(z) - Delta^(m-1)(z-1)]*/while poly(expand(p(z) - p(z-1)), [z]) poly(0, [z]) dod:= d + 1;p:= poly(expand(p(z) - p(z-1)), [z]);pList[d]:= p;end_while;/* Compute the unique lexicographic ideal L associated to theHilbert polynomial ’pList[d]’ in R = K[x_0,...,x_{n-d+1}]. */L:= compute_L_p(pList[d], n-d+1);/* Use procedure ’compute_ideals’ to compute all saturated stableideals with the same Hilbert polynomial as L in R.From now on we collect all computed ideals in the set M. */M:= compute_ideals(L, n-d+1);d:= d-1;/* Treat the special cases for constant and linear Hilbert polynomials:deg = 0 -- the Hilbert polynomial is constant. Hence, the doublesaturation of all ideals computed equals the wholepolynomial ring.deg = 1 -- the Hilbert polynomial is linear. Thus, the set Mcontains all possible double saturations of the idealsassociated to the Hilbert polynomial at this time.The case with deg > 1 is treated below within the for-loop */if deg = 0 and args(0) = 3 and args(3) = hold(All) thenprint(Unquoted, "constant Hilbert polynomial: the double saturation");
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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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University of Paderborn Department of Mathematics Diploma Thesis ...

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