University of Paderborn Department of Mathematics Diploma Thesis ...

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120 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSThe advantage of the above algorithm is that it can be used to compute single valuesh R/I (i) of a Hilbert function for large i ∈ N efficiently. The algorithm stated in Remark4.25 would have to compute h R/I (j) for all j of i is largerthan the integer k of Lemma 4.26. If this is the case, there is no need to compute all valuesh R/I (j) for j < i. It suffices to compute the Hilbert polynomial, from which one obtainsh R/I (i). Next we give the source code of a possible implementation of the above algorithmin MuPAD.MuPAD Source Code 4.28. Let H = H(t) be the numerator of a reduced Hilbert seriesof R/I for I ⊂ R a saturated stable ideal. Let d be the exponent of (1 − t) in the denominatorof H and n the index of the last variable in R. The procedure below computes thevalues of the corresponding Hilbert function up to an upper bound, which has to be givenas the third argument for the procedure.Input for the procedure compute Hilbert function.◦ H — the numerator of a reduced Hilbert series◦ d — the power of (1 − t) in the denominator of the Hilbert series◦ upper bound — the value, i.e. the upper bound, up to which the values of the Hilbertfunction (associated to the given Hilbert series) should be computed◦ n — the index of the last variable of the polynomial ring ROutput of the procedure compute Hilbert function.◦ values — a list containing the values of the Hilbert functioncompute_Hilbert_function:= proc(H, d, upper_bound, n)local coeffs, values, i, j, p, L_p, lastMon, G_bound;begincoeffs:= [coeff(poly(expand(1-t)^d, [t]))]:values:= [0 $ upper_bound + 1]:values[1]:= 1:p:= compute_Hilbert_polynomial2(H, d):L_p:= compute_L_p(p, n):lastMon:= L_p[nops(L_p)]:G_bound:= _plus(lastMon[i] $ i = 1..n+1):for i from 2 to d+1 dovalues[i]:= coeff(H,i-1):for j from 1 to i-1 dovalues[i]:= values[i] - (values[i-j] * coeffs[d+1-j])
4.7. COMPUTING ALL HILBERT FUNCTIONS TO A HILBERT POLYNOMIAL 121end_for:end_for:for i from d+2 to G_bound - 1 dovalues[i]:= coeff(H,i-1):for j from 1 to d dovalues[i]:= values[i] - (values[i-j] * coeffs[d+1-j])end_for:end_for:/* Now compute the Hilbert function with the Hilbert polynomial. */for i from G_bound to upper_bound+1 dovalues[i]:= p(i-1):end_for:return(values);end_proc:Example 4.29. We determine the values of the Hilbert functions associated to the Hilbertseries computed in Example 4.24.MuPAD>> compute_Hilbert_function(poly(-t^7+t^4+t^3+t^2+t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^4+t^3+t^2+2*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-2*t^6+t^5+t^4+t^3+t^2+t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^3+2*t^2+2*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^6+t^4+t^3+2*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^5+t^4+t^3-t^2+3*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(3*t+1,[t]),2,10,4);>> compute_Hilbert_function(poly(-t^5+t^4+t^2+2*t+1,[t]),2,10,4);Output[1, 3, 6, 10, 15, 20, 25, 29, 33, 37, 41][1, 4, 8, 13, 17, 21, 25, 29, 33, 37, 41][1, 3, 6, 10, 15, 21, 25, 29, 33, 37, 41][1, 4, 9, 13, 17, 21, 25, 29, 33, 37, 41][1, 4, 7, 11, 16, 21, 25, 29, 33, 37, 41][1, 5, 8, 12, 17, 21, 25, 29, 33, 37, 41][1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41][1, 4, 8, 12, 17, 21, 25, 29, 33, 37, 41]
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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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