University of Paderborn Department of Mathematics Diploma Thesis ...

124 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSThe ideal L k is the unique lexicographic ideal associated to the Hilbert polynomial p k (z) :=13 z3 + 5 2 z2 + 7 z + (k − 1), k = 3, 4, 5, 6, since we have6p R/L3 (z) = 1 3 z3 + 5 2 z2 + 7 6 z + 2,p R/L4 (z) = 1 3 z3 + 5 2 z2 + 7 6 z + 3,p R/L5 (z) = 1 3 z3 + 5 2 z2 + 7 6 z + 4,p R/L6 (z) = 1 3 z3 + 5 2 z2 + 7 6 z + 5.Again, L j+1 is obtained from L j , j = 3, 4, 5, by the expansion of the last monomial generatorof L j . As above, we sum up the results in a table:number of saturated number of doublestable ideals saturationsk = 3 23 3k = 4 51 3k = 5 106 3k = 6 212 3The first interesting observation is that the number of all saturated stable ideals grows fastif one slightly increases the exponent of the last variable of the last monomial generator inthe lexicographic ideal considered. Hence, it is quite easy to find concrete examples, wherethe number of all saturated stable ideals to a given Hilbert polynomial is too large to becomputed within a few minutes or even within a few hours on a usual computer. Thisseems to be clear if one increases the number of variables of the polynomial ring and thedegree of the Hilbert polynomial considered, but the above examples suggest that we donot necessarily have to use a large number of variables and a Hilbert polynomial of highdegree – even for linear Hilbert polynomials or Hilbert polynomials of degree 3, there maybe a large number of saturated stable ideals to a given Hilbert polynomial.A rather informal explanation of this phenomenon may be found if we have a closer look athow Algorithm 3.45 computes all saturated stable ideals to a given Hilbert polynomial. Themost important “algorithmic tool” within the procedure is the performance of expansionsand contractions of monomials. In Chapter 3, we saw that we only have to contract thosemonomials, which contain last variable x 3 respectively x 4 . Since, among other computationalsteps, the main part of Algorithm 3.45 is to compute all possible contractions andexpansions in a certain ideal such that one obtains saturated stable ideals with the sameHilbert polynomial, it becomes obvious that the number of all such possible expansionsand contractions strongly depends on the exponent of x 3 respectively x 4 in the monomialgenerators of the ideal considered. If we start with a saturated stable ideal, which has only