University of Paderborn Department of Mathematics Diploma Thesis ...

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114 CHAPTER 4. ALGORITHMS FOR STABLE IDEALS4.6 Computing all Hilbert series to a given HilbertpolynomialIn Chapter 2, Theorem 2.21 states that there is a unique lexicographic ideal L H to anyHilbert series H R/I (t) of R/I for I ⊂ R a saturated homogeneous ideal. We stated a pseudocode algorithm to compute all Hilbert series to a given Hilbert polynomial in Chapter 3,Algorithm 3.39. There we used the unique lexicographic ideal L p associated to p(z) to findall lexicographic ideals L H associated to Hilbert series H(t), such that p R/LH (z) = p(z).By Corollary 3.37, the lexicographic ideals to any Hilbert series associated to p(z) can becomputed by procedure 4.17. Thus, we can use this procedure to compute all such lexicographicideals to all possible Hilbert series associated to the given Hilbert polynomialp(z). Then we find all such possible Hilbert series by simply computing the Hilbert seriesof any of the ideals found by procedure 4.17.An implementation of this algorithm is listed below.MuPAD Source Code 4.23. Let p be the Hilbert polynomial of R/I for I ⊂ R asaturated stable ideal and n the index of the last variable of R. The following procedureexpects two or three arguments.By using it with only the two arguments p and n, it computes the numerators of all reducedHilbert series associated to p. If one uses the additional third argument ’All’, it doesnot only compute the reduced, but also the non-reduced numerators of the Hilbert seriesassociated to p.Input for the procedure compute all Hilbert series.◦ 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 Hilbert series.◦ Hilbert series — a set of polynomials, encoding the numerators of all Hilbert series(in reduced representation) associated to p. If the additional third input parameterAll is used, then the procedure also returns the numerators of the non-reducedHilbert series. In this case, Hilbert series is a set of lists, where each list containsas first entry the non-reduced, and as second entry the reduced representation of theHilbert series associated to p.compute_all_Hilbert_series:= proc(p, n /*, All*/)local L_p, Ideals, Hilbert_series, i;
4.6. COMPUTING ALL HILBERT SERIES TO A HILBERT POLYNOMIAL 115begin/* Use procedure ’compute_L_p’ to compute the unique lexicographicideal L_p associated to the Hilbert polynomial p. */L_p:= compute_L_p(p, n);/* Use procedure ’compute_ideals’ to compute all saturatedstable ideals with the same Hilbert polynomial and thesame double saturation as L_p. */Ideals:= compute_ideals(L_p, n);/* Use procedure ’compute_Hilbert_series’ to compute all possibleHilbert series. If the third argument ’All’ is used, then computethe numerators of the reduced and the non-reduced Hilbert seriesfor each ideal. Otherwise, only compute the reduced numerators. */Hilbert_series:= [0 $ nops(Ideals)];if args(0) = 3 and args(3) = hold(All) thenfor i from 1 to nops(Ideals) doHilbert_series[i]:= [compute_Hilbert_series(Ideals[i], n)];end_for;elsefor i from 1 to nops(Ideals) doHilbert_series[i]:= op(compute_Hilbert_series(Ideals[i], n),2);end_for;end_if;return({op(Hilbert_series)});end_proc:Example 4.24. Again, let us consider p(z) = 4z + 1 and R := K[x 0 , x 1 , x 2 , x 3 , x 4 ]. Wewant to compute the numerators of all reduced Hilbert series H(t) associated to the Hilbertpolynomial p(z).Note that we do not use the additional third parameter ’All’ in the call of the procedurecompute all Hilbert series below. In general, it suffices to know the numeratorsof any of the reduced Hilbert series associated to p(z).
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University of PaderbornDepartment o
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”Im großen Garten der Geometriek
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