University of Paderborn Department of Mathematics Diploma Thesis ...

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.).