University of Paderborn Department of Mathematics Diploma Thesis ...

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44 CHAPTER 2. STABLE IDEALSBinomial representation: Polynomial representation: Sum of former polynomials:( ) z + 2p 1 (z) :=p 1 (z) = 1 22 z2 + 3 1∑2 z + 1 p i (z) = 1 2 z2 + 3 2 z + 1( ) z + 2 − 1p 2 (z) :=2( ) z + 2 − 2p 3 (z) :=2( ) z + 2 − 3p 4 (z) :=2( ) z + 1 − 4p 5 (z) :=1( ) z + 1 − 5p 6 (z) :=1( ) z + 0 − 6p 7 (z) :=0( ) z + 0 − 7p 8 (z) :=0( ) z + 0 − 8p 9 (z) :=0( ) z + 0 − 9p 10 (z) :=0( )z + 0 − 10p 11 (z) :=0( )z + 0 − 11p 12 (z) :=0p 2 (z) = 1 2 z2 + 1 2 zp 3 (z) = 1 2 z2 − 1 2 zp 4 (z) = 1 2 z2 − 3 2 z + 1p 5 (z) = z − 3p 6 (z) = z − 4p 7 (z) = 1p 8 (z) = 1p 9 (z) = 1p 10 (z) = 1p 11 (z) = 1p 12 (z) = 1i=12∑p i (z) = z 2 + 2z + 1i=13∑p i (z) = 3 2 z2 + 3 2 z + 1i=14∑p i (z) = 2z 2 + 2i=15∑p i (z) = 2z 2 + z − 1i=16∑p i (z) = 2z 2 + 2z − 5i=17∑p i (z) = 2z 2 + 2z − 4i=18∑p i (z) = 2z 2 + 2z − 3i=19∑p i (z) = 2z 2 + 2z − 2i=1∑10i=1∑11i=1∑12i=1p i (z) = 2z 2 + 2z − 1p i (z) = 2z 2 + 2zp i (z) = 2z 2 + 2z + 1This provides a 1 = a 2 = a 3 = a 4 = 2, a 5 = a 6 = 1 and a 7 = a 8 = a 9 = a 10 = a 11 = a 12 = 0in the terminology of Theorem 2.32. Hence, the upper bound for the degrees is 12. Indeed,the ideal L p := (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 6 3) ⊂ R contains one monomial generator of degree 12and we obtain p R/Lp (z) = 2z 2 + 2z + 1.
Chapter 3Operations on stable idealsJust like in the preceding chapters we will only consider stable ideals in R := K[x 0 , . . . , x n ].In this chapter we want to present some useful algorithmic tools, which we will apply to theset of minimal generators of given saturated stable ideals. With the help of these “tools”,we prove one of the main results of this thesis, i.e. a characterization of all saturated stableideals with the same double saturation and the same Hilbert polynomial due to [16]. Wewill see that all these ideals are linked by finite steps of so-called contractions and expansions.In particular, given two saturated stable ideals I, J ⊂ R with the same Hilbert polynomialand the same double saturation, we will be able to compute J from I and vice versa bypairs of contractions and expansions. The proof of this assertion will be constructive, suchthat it provides an algorithm, which computes all saturated stable ideals with the samedouble saturation to a given Hilbert polynomial.The last section of the chapter is dedicated to prove a similar result for all saturated stableideals with the same Hilbert polynomial without the assumption that these ideals have thesame double saturation. As a consequence of this result, we give a pseudo code versionof the algorithm to compute all saturated stable with the same Hilbert polynomial. Theimplementation of all algorithms presented within this chapter can be found in Chapter 4.3.1 Left-shifts and right-shifts of monomialsWithin this section, we introduce the definitions of so-called left-shifts and right-shifts ofmonomials. They will help to define contractions and expansions of monomials in the nextsection. These contractions and expansions will not only play an important role in provingthe central results in the following sections of this chapter, but also in the algorithm tocompute all saturated stable ideals to a given Hilbert polynomial.In the following, we will often use capital letters as exponents of x. In case we write x A , Ashall always denote an (1 × (n + 1)) exponent vector with components a 0 , a 1 , . . . , a n ∈ N 0 .45
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University of Paderborn Department of Mathematics Diploma Thesis ...

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