University of Paderborn Department of Mathematics Diploma Thesis ...

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82 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSMuPAD>> compute_Hilbert_polynomial([[1,0,0,0,0],[0,5,0,0,0],[0,4,3,0,0],>> [0,4,2,6,0]], 4);Output2poly(2 z+ 2 z + 1, [z])i.e. p R/I4 (z) = 2z 2 + 2z + 1, as expected, since I 4 is the lexicographic ideal L p associatedto the Hilbert polynomial p(z) = 2z 2 + 2z + 1 by Theorem 2.25.Later, we need an algorithm, which computes the Hilbert polynomial from a given reducedHilbert series. Thus, we also give a slightly different version of the source code presentedin 4.3.MuPAD Source Code 4.5. The procedure below expects as input the numerator of areduced Hilbert series and the power d of (1 − t) in its denominator. It computes theHilbert polynomial associated to the numerator H of the Hilbert series.Input for the procedure compute Hilbert polynomial2.◦ H — the numerator of a reduced Hilbert series◦ d — the power of (1 − t) in the denominator of the Hilbert seriesOutput of the procedure compute Hilbert polynomial2.◦ p — the Hilbert polynomial of R/I, where I is the ideal generated by the monomialsin Mcompute_Hilbert_polynomial2:= proc(H, d)local p, j;begin/* H is the numerator of a reduced Hilbert series andd the power of (1-t) in the denominator of the Hilbertseries. */p:= poly(0, [z]);for j from 0 to degree(H) dop:= p + poly(expand(coeff(H,j)*binomial(z+d-1-j,d-1)),[z]);end_for;return(p);end_proc:
4.3. COMPUTING THE LEXICOGRAPHIC IDEAL L P 83Since the code is nearly identical to that of 4.3, we do not consider any examples to demonstrateits use – an application of 4.5 is given in a later section, where we compute Hilbertfunctions.The computation of the lexicographic ideal L p to a given Hilbert polynomial p(z) is theaim of the next section.4.3 Computing the lexicographic ideal L pIn Theorem 2.25, we defined the unique lexicographic L p to a given Hilbert polynomial p(z)such that p(z) = p R/Lp (z). By Remark 1.11, there are integers m 0 ≥ m 1 ≥ . . . ≥ m deg p ≥ 0such that we can write p(z) in the formp(z) = g(m 0 , . . . , m deg p ; z) :=deg p∑i=0( ) z + i−i + 1( z + i − mii + 1From m 0 , . . . , m deg p , we compute the values of the integers a 0 , . . . , a r , such that, by Theorem2.25, L p is generated minimally by the set{x 0 , x 1 , . . . , x n−r−2 ,x ar+1n−r−1,x arn−r−1 · x a r−1+1n−r ,x arn−r−1 · x a r−1n−r · x a r−2+1n−r+1 , . . . ,x arn−r−1 · x a r−1n−r · x a r−2n−r+1 · . . . · x a 2n−3 · x a 1+1n−2 ,x arn−r−1 · x a r−1n−r · x a r−2n−r+1 · . . . · x a 1n−2 · x a 0n−1}and a r = m r , a r−1 = m r−1 − m r , . . . , a 0 = m 0 − m 1 .The following algorithm (in pseudo code) computes the lexicographic ideal L p to a givenHilbert polynomial p(z).Algorithm 4.6. (Computation of the lexicographic ideal L p ) Let p(z) be the Hilbert polynomialof R/I for I ⊂ R a homogeneous saturated ideal. First we describe the algorithmin pseudo code to illustrate the ideas used. Then we present a possible MuPAD implementation.Input. p(z), a Hilbert polynomial and n, the index of the last variable of R.(i) Compute the values of m 0 , . . . , m deg p by Remark 1.11.(ii) From m 0 , . . . , m deg p compute the values of a 0 , . . . , a deg p with respect to Theorem 2.25.(iii) Compute the set of minimal generators of L p by Theorem 2.25.).
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University of PaderbornDepartment o
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”Im großen Garten der Geometriek
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AbstractThe main result of this the
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Glossary of MuPAD procedures 129Bib
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This thesis is organized in four ch
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6described in Chapter 4.Additionall
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University of Paderborn Department of Mathematics Diploma Thesis ...

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