University of Paderborn Department of Mathematics Diploma Thesis ...

4.5. COMPUTING STABLE IDEALS TO A GIVEN HILBERT POLYNOMIAL 109These four ideals are given by (x 2 0, x 0 x 1 , x 0 x 2 , x 0 x 3 , x 2 1, x 1 x 2 , x 3 2), (x 2 0, x 0 x 1 , x 0 x 2 , x 2 1, x 1 x 2 , x 2 2),(x 0 , x 2 1, x 1 x 2 2, x 1 x 2 x 3 , x 3 2), (x 0 , x 2 1, x 1 x 2 , x 4 2, x 3 2x 3 ). Hence, the different double saturations,which appear among all saturated stable ideals to the Hilbert polynomial p(z) = 4z + 1,are (x 0 , x 1 , x 4 2) (this is the double saturation of the unique lexicographic ideal associatedto p(z)), (x 0 , x 2 1, x 1 x 2 , x 3 2) and (x 2 0, x 0 x 1 , x 0 x 2 , x 2 1, x 1 x 2 , x 2 2) (these two double saturationsbelong to the four ideals stated above, whose double saturation differs from that of theunique lexicographic ideal). We can compute these four ideals as follows:MuPAD>> M:= compute_all_ideals(poly(4*z + 1, [z]), 4, All):>> M[2];Output{[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 4, 0]],[[1, 0, 0, 0], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 3, 0]],[[2, 0, 0, 0], [1, 1, 0, 0], [1, 0, 1, 0], [0, 2, 0, 0], [0, 1, 1, 0],[0, 0, 2, 0]]}>> M[3];Output3The set of lists M[2] encodes exactly those double saturations stated above. The valueM[3] gives us the number of elements of the set M[2], i.e. the number of all differentdouble saturations appearing among all saturated stable ideals associated to the Hilbertpolynomial p(z).Example 4.22. Consider p(z) := 2z 2 + 2z + 1 and R := K[x 0 , x 1 , x 2 , x 3 , x 4 ]. In Example4.19, we saw that there are 77 saturated stable ideals with the same Hilbert polynomialand the same double saturation as the unique lexicographic ideal L p .Now we compute the number of all saturated stable ideals to the given Hilbert polynomial:MuPAD>> nops(compute_all_ideals(poly(2*z^2 + 2*z + 1, [z]), 4));Output84i.e. there are 84 saturated stable ideals to the given Hilbert polynomial. Hence, among