University of Paderborn Department of Mathematics Diploma Thesis ...

76 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSm 1 > m 2 > . . . > m r with respect to the lexicographic order, d i := deg m i , and letl i ∈ {0, . . . , n} be the index of the last variable appearing in m i . Then, by Theorem 2.17,the Hilbert series of R/I is given byH R/I (t) = 1 − ∑ ri=1 (1 − t)li · t d i(1 − t) n+1 .We will see below that the implementation of an algorithm, which computes the Hilbertseries H R/I (t) from the set of minimal generators of I goes straight forward.In the following source code we use lists to represent monomials, i.e. the monomialhas to be identified with the listx a 00 · x a 11 · . . . · x a n−1n−1 · x ann ∈ M[a 0 , a 1 , . . . , a n−1 , a n ].Then, the set of generators of any stable ideal can be encoded by a list of such lists.Note that (in our special situation) it is not a good choice, to encode the set of generatorsof an ideal by a set of lists instead of using a list of lists, because the elements of a set arenot ordered in MuPAD. Hence, it would be difficult to keep the monomials in lexicographicorder, which is necessary for most of the algorithms presented in the course of this chapter.The set of generators of any stable ideal appearing in this chapter will always be orderedwith respect to the lexicographic order.MuPAD Source Code 4.1. The MuPAD procedure below expects a list of lists encodingthe set of minimal generators of a stable ideal I and the index n of the last variable ofR = K[x 0 , . . . , x n ]. It returns the non-reduced and the reduced Hilbert series of R/I.Input for the procedure compute Hilbert series.◦ M — a list of lists encoding the set of generators of some stable ideal in lexicographicorder◦ n — the index of the last variable of the polynomial ring ROutput of the procedure compute Hilbert series.◦ H — the numerator of the non-reduced Hilbert series◦ H red — the numerator of the reduced Hilbert series