University of Paderborn Department of Mathematics Diploma Thesis ...

4.8. SOME CONCLUSIONS AND EXPERIMENTAL RESULTS 125one monomial generator with the last variable x 3 respectively x 4 and the exponent of x 3respectively x 4 is small, say 4 or 5, as in our examples above, one cannot perform as manypossible contractions and expansions as if the power of x 3 respectively x 4 is large. Byincreasing the exponent of x 3 respectively x 4 by 1, we simultaneously increase the numberof possible expansions and contractions, which can be performed.The second interesting observation is that the number of different double saturations in theabove examples is independent of the value of k, k = 3, 4, 5, 6. The reason seems to be thata different value for k does not change the double saturation of the unique lexicographicideals considered: We haveandsat x3 ,x 4((x0 , x 1 , x 5 2, x 4 2x k 3) ) = (x 0 , x 1 , x 4 2)sat x4 ,x 5((x0 , x 3 1, x 2 1x 3 2, x 2 1x 2 2x 2 3, x 2 1x 2 2x 3 x k 4) ) = (x 0 , x 3 1, x 2 1x 3 2, x 2 1x 2 2x 3 )for all k ∈ {3, 4, 5, 6}. Hence, the number of different double saturations seems to dependon the exponent vectors of the other monomial generators of the lexicographic ideal L knot containing the last variable x 3 respectively x 4 . By increasing the value of k, we do notaffect these generators at all. This could be an explanation for the fact that the numberof double saturations remains the same (of course, these results strongly depend on thechoice of our examples).To summarize, we point out that Algorithm 3.45 and its implementation by the proceduredescribed in 4.20 can be used, to investigate the effect of the structure of the lexicographicideal L p associated to some Hilbert polynomial p(z) on the number of different saturatedstable ideals with the same Hilbert polynomial and even on the number of different doublesaturations appearing among these ideals. A lot of the other algorithms stated withinChapter 4, which finally lead to the procedure in 4.20, are not only useful in the contextof the algorithm to determine all saturated stable ideals with a given Hilbert polynomial:e.g. we have given algorithms to compute Hilbert series, Hilbert polynomials and evenHilbert functions of saturated stable ideals in a rather efficient way. Note that there areless computer algebra systems today (even special purpose systems for the field of computationalcommutative algebra), which provide procedures for the explicit computation ofHilbert polynomials or the values of the Hilbert function (these systems often restrict itsusers to the computation of Hilbert series). Additionally, the algorithm to compute thelexicographic ideal L p is a useful tool, since one can easily compute an ideal I (which is alsostable, saturated and lexicographic) such that p R/I (z) = p(z) for an arbitrary Hilbert polynomialp(z). Together with the characterization of this special ideal presented in Corollary3.40, the procedures in Chapter 4 leave much freedom for several more experiments withstable ideals.