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122 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSThe first list gives the values h R/Lp (i), 0 ≤ i ≤ 9, of the Hilbert function of R/L p for theunique lexicographic ideal L p associated to p(z). As we see now, the above values demonstratethe result presented in Corollary 3.40, where we characterized the ideal L p as theunique lexicographic ideal, such that the values of the Hilbert function h R/Lp are minimalamong the values of all Hilbert functions associated to p(z).Note that we have determined the first ten values of all Hilbert functions associatedto the Hilbert polynomial p(z) = 4z + 1 (the underlying polynomial ring is given byR := K[x 0 , x 1 , x 2 , x 3 , x 4 ]). This is the case, since in Example 4.24 we computed all Hilbertseries associated to p(z). Hence, there cannot be any other Hilbert function than those,whose first ten values are presented above.As this particular example shows, there also is a unique lexicographic ideal L H associatedto some Hilbert series H(t), such that p R/LH (z) = p(z) and the values of the Hilbertfunction h R/LH are maximal in each degree (compared to the values of all other Hilbertfunctions associated to p(z) in a fixed degree): The second line from below gives the firstten values of this Hilbert function. In each degree, the value of this Hilbert function isat least as large as the corresponding value of any other Hilbert function associated to p(z).In the last section of this chapter we state some interesting experimental results and conclusionsconcerning the “behaviour” of Algorithm 3.45 to compute all saturated stableideals to a given Hilbert polynomial p(z) and the effect of the structure of the uniquelexicographic ideal L p associated to p(z) by Theorem 2.25.4.8 Appendix: Some conclusions and experimentalresultsWithin the time, the algorithms have been implemented in MuPAD, several more exampleshave been computed. These examples could not be discussed in detail in the precedingsections, since this would have overstepped the framework of this thesis. Nevertheless,some of the examples, which were originally computed for checking the correctness of theimplementations presented here, provided some interesting results (most of the examplescan be found in the file “testfile.mu”, a special file for checking the MuPAD proceduresof this chapter automatically, which is also contained on the CD in the appendix to thisthesis).It turned out that the “structure” of the unique lexicographic ideal L p associated to someHilbert polynomial p(z) (see Theorem 2.25) has a strong effect not only on the number ofall saturated stable ideals I ⊂ K[x 0 , . . . , x n ] with p(z) = p R/I (z), but also on the numberof different double saturations appearing among these ideals.As a first example, let us consider the four ideals defined by L k := (x 0 , x 1 , x 5 2, x 4 2x k 3) in the
4.8. SOME CONCLUSIONS AND EXPERIMENTAL RESULTS 123polynomial ring R := K[x 0 , x 1 , x 2 , x 3 , x 4 ] for k = 3, 4, 5, 6, i.e.L 3 = (x 0 , x 1 , x 5 2, x 4 2x 3 3),L 4 = (x 0 , x 1 , x 5 2, x 4 2x 4 3),L 5 = (x 0 , x 1 , x 5 2, x 4 2x 5 3),L 6 = (x 0 , x 1 , x 5 2, x 4 2x 6 3).These ideals are the unique lexicographic ideals (see Theorem 2.25) to the Hilbert polynomialsp k (z) = 4z + (k − 2), k = 3, 4, 5, 6, i.e.p R/L3 (z) = 4z + 1,p R/L4 (z) = 4z + 2,p R/L5 (z) = 4z + 3,p R/L6 (z) = 4z + 4.Note that p R/L3 (z) = 4z + 1 is the Hilbert polynomial we already considered in Example4.18 and which is also discussed by Alyson Reeves in [16]. Furthermore, note that we obtainL j+1 from L j , j = 3, 4, 5, by performing one expansion of the last monomial generator of L j .We want to compute the number of all saturated stable ideals I ⊂ R with p R/I (z) = p k (z),k = 3, 4, 5, 6, and the number of different double saturations appearing among these ideals.Using the procedure compute all ideals stated in 4.20, we obtain:number of saturated number of doublestable ideals saturationsk = 3 12 3k = 4 23 3k = 5 43 3k = 6 79 3It is interesting to see that the number of all saturated stable ideals to the given Hilbertpolynomial increases fast, whereas the number of different double saturations among theseideals remains the same.Before we try to explain these observations, we consider another example: Let L k :=(x 0 , x 3 1, x 2 1x 3 2, x 2 1x 2 2x 2 3, x 2 1x 2 2x 3 x k 4) for k = 3, 4, 5, 6 in R := K[x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ], i.e.L 3 = (x 0 , x 3 1, x 2 1x 3 2, x 2 1x 2 2x 2 3, x 2 1x 2 2x 3 x 3 4),L 4 = (x 0 , x 3 1, x 2 1x 3 2, x 2 1x 2 2x 2 3, x 2 1x 2 2x 3 x 4 4),L 5 = (x 0 , x 3 1, x 2 1x 3 2, x 2 1x 2 2x 2 3, x 2 1x 2 2x 3 x 5 4),L 6 = (x 0 , x 3 1, x 2 1x 3 2, x 2 1x 2 2x 2 3, x 2 1x 2 2x 3 x 6 4).
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University of PaderbornDepartment o
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”Im großen Garten der Geometriek
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Glossary of MuPAD procedures 129Bib
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University of Paderborn Department of Mathematics Diploma Thesis ...

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