University of Paderborn Department of Mathematics Diploma Thesis ...

4.5. COMPUTING STABLE IDEALS TO A GIVEN HILBERT POLYNOMIAL 113⎧1 if j = 05 if j = 112 if j = 223 if j = 3⎪⎨39 if j = 4h R/I4 (j) =59 if j = 583 if j = 6111 if j = 7143 if j = 8⎪⎩179 if j = 9⎧−1 if z = 03 if z = 111 if z = 223 if z = 3⎪⎨39 if z = 4p R/I4 (z) =59 if z = 583 if z = 6111 if z = 7143 if z = 8⎪⎩179 if z = 9(Later within 4.28 we present a procedure to compute the values of the Hilbert function toa given reduced Hilbert series of R/I for I ⊂ R = K[x 0 , . . . , x n ] a saturated homogeneousideal. This procedure has been used to compute the values of the Hilbert functions above.)It is amazing that the values h R/I1 (j) and h R/I3 (j), j = 0, 1, . . . , 9, equal those of thecorresponding Hilbert polynomial p R/I1 (z) = p R/I3 (z), z = 0, 1, . . . , 9. In the case of R/I 2we have h R/I2 (j) ≠ p R/I2 (j) for j = 0, 1, 2, 3 and h R/I2 (j) = p R/I2 (j) for j = 4, 5, . . . , 9.Finally for R/I 4 we obtain h R/I4 (j) ≠ p R/I4 (j) for j = 0, 1, 2 and h R/I4 (j) = p R/I4 (j) forj = 3, 4, . . . , 9.Comparing the values of the Hilbert functions of R/I 1 , R/I 2 , R/I 3 and R/I 4 in a fixeddegree j ∈ {0, 1, 2, . . . , 9}, we can see that we haveh R/I2 (j) ≤ h R/I4 (j) ≤ h R/I1 (j) = h R/I3 (j)for all j and even h R/I2 (j) < h R/I4 (j) for j ∈ {1, 2, . . . , 9}. Hence, we have a uniquely determinedminimal Hilbert function h R/I2 among all the double saturations I 1 , I 2 , I 3 and I 4appearing among the saturated stable ideals to the Hilbert polynomial p(z) = 2z 2 + 2z + 1.This minimal Hilbert function belongs to R/I 2 , where I 2 is the double saturation of theunique lexicographic ideal L p associated to the Hilbert polynomial p(z). This agrees withthe characterization of the ideal L p in Corollary 3.40 (ii).From this special example we additionally conclude that there may not be a uniquelydetermined ideal I with maximal Hilbert function of R/I, where I is a double saturationappearing among all saturated stable ideals to a given Hilbert polynomial. In our case,the Hilbert functions of R/I 1 and R/I 3 are both maximal in the sense of our comparisonabove.In the next section we present the source code of a procedure to compute all Hilbert seriesassociated to a given Hilbert polynomial.