University of Paderborn Department of Mathematics Diploma Thesis ...

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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
4.1. COMPUTING HILBERT SERIES OF STABLE IDEALS 77compute_Hilbert_series:= proc(M, n)local di, li, monom, H, H_red, i, j;beginH:= poly(1,[t]);for i from 1 to nops(M) domonom:= M[i];for j from n+1 downto 1 do/* Determine index li of last variable in monom */if monom[j] 0 thenli:= j-1;break;end_if;end_for;/* Determine the degree of monom */di:= _plus(op(monom));H:= H - poly((1-t)^li * t^di,[t]);end_for;/* At this point H is the numerator of the non-reducedHilbert series. Now compute the numerator H_red ofthe reduced Hilbert series via gcd. */H_red:= H/gcd(H, poly((1-t)^(n+1), [t]));if H_red(0) = -1 thenH_red:= H_red * poly(-1, [t]);end_if;return(H, H_red);end_proc:To illustrate how to use the procedure, we consider a few examples where we compute theHilbert series of some stable ideals, which also occurred in some of the examples of thepreceding two chapters.Example 4.2. First, let R := K[x 0 , x 1 , x 2 ], i.e. n = 2. We will consider some “larger”examples below in (iii) and (iv).(i) Consider the stable ideal I 1 := (x 2 0, x 0 x 1 , x 2 1) ⊂ R. The set of generators is encoded bythe list of lists[[2, 0, 0], [1, 1, 0], [0, 2, 0]].The value of n is 2. Hence, we compute the Hilbert series of R/I 1 by the following simplecommand:
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University of PaderbornDepartment o
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University of Paderborn Department of Mathematics Diploma Thesis ...

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