University of Paderborn Department of Mathematics Diploma Thesis ...

72 CHAPTER 3. OPERATIONS ON STABLE IDEALSandR/(J + x n R) ∼ = (R/x n R)/((J + x n R)/x n R) ∼ =¯R/ ¯J,we conclude p ¯R/ Ī(z) = p R/(I+xnR)(z) = p R/I (z) − p R/I (z − 1) = p R/J (z) − p R/J (z − 1) =p R/(J+xnR)(z) = p ¯R/ ¯J(z). This proves the claim.It follows p ¯R/ satxn−1 (Ī) (z) = p ¯R/ satxn−1 ( ¯J)(z). Since we have deg p ¯R/ satxn−1 (Ī) (z) = d − 1, andsat (Ī), sat xn−1 x n−1( ¯J) are saturated stable ideals in ¯R, we conclude by induction that thereis a finite sequence of contractions and expansions to compute sat xn−1 ( ¯J) from sat (Ī) xn−1(note that the expansions in ¯R are not the same as the expansions in R).Now view sat (Ī) and sat xn−1 x n−1( ¯J) as ideals in R. Because the computation of the saturationof every stable ideal in ¯R corresponds to the computation of the double saturation ofthe same ideal viewed as a saturated stable ideal in R, we obtain sat (Ī) = sat xn−1 x n−1 ,x n(I)and sat xn−1 ( ¯J) = sat xn−1 ,x n(J) in the ring R. By Lemma 3.17, there are sequences ofcontractions, which take I to sat xn−1 ,x n(I) and J to sat xn−1 ,x n(J). By our induction hypothesis,we can first compute sat xn−1 ,x n(I) from I by a sequence of contractions, thencompute sat xn−1 ,x n(J) from sat xn−1 ,x n(I) (again viewed as saturated ideals in ¯R) and finallycompute J from sat xn−1 ,x n(J) by expansions (recall that we can interpret the sequence ofcontractions taking J to sat xn−1 ,x n(J) as a sequence of expansions taking sat xn−1 ,x n(J) toJ, since the contractions performed always involve monomials with the last variable x n−1 ).This completes the proof.Remark 3.44. Note that the preceding characterization of all saturated stable ideals withthe same Hilbert polynomial is weaker, than our characterization of all saturated stableideals with the same double saturation and the same Hilbert polynomial. Theorem 3.32states that the latter ideals are linked by finite sequences of paired contractions and expansion.In the more general situation of computing all saturated stable ideal with thesame Hilbert polynomial, we cannot restrict ourselves to performing such pairs of contractionsand expansions anymore. As the above proof suggests, we may need some additionalexpansions to compute all these ideals. Nevertheless, we can still use contractions andexpansions as algorithmic tools to reach our aim.In Chapter 4 we give the source code of an algorithm for the construction of all saturatedstable ideals with the same Hilbert polynomial. Now, we use the proof of Theorem 3.43 togive a pseudo code version of this algorithm.Algorithm 3.45. (Computation of all saturated stable ideals to a given Hilbert polynomial)Let p(z) ≠ 0, d := deg p(z) be the Hilbert polynomial of R/I for I ⊂ R a saturatedstable ideal. In the following, we denote by R (i) , 1 ≤ i ≤ n, the polynomial ringK[x 0 , x 1 , . . . , x n−i ].Input. A Hilbert polynomial p(z) ≠ 0.1. Compute ∆p(z) := p(z) − p(z − 1), ∆ 2 p(z) := ∆p(z) − ∆p(z − 1), . . . , ∆ d p(z) :=∆ d−1 p(z) − ∆ d−1 p(z − 1). Then ∆ d p(z) = c for some c ∈ N.