University of Paderborn Department of Mathematics Diploma Thesis ...

4.6. COMPUTING ALL HILBERT SERIES TO A HILBERT POLYNOMIAL 1174 3 2poly(- t + t + t + 2 t + 1, [t])],8 7 6 4 3 2[poly(t - 4 t + 5 t - 8 t + 12 t - 7 t + 1, [t]),5 4 3 2poly(- t + t + t - t + 3 t + 1, [t])],8 7 6 5 4 3 2[poly(t - 4 t + 6 t - 5 t + 2 t + 2 t - 2 t - t + 1, [t]),5 4 2poly(- t + t + t + 2 t + 1, [t])],9 8 7 6 4 3 2[poly(t - 3 t + 2 t + t - 4 t + 6 t - 3 t - t + 1, [t]),6 4 3poly(- t + t + t + 2 t + 1, [t])],9 8 7 6 2[poly(2 t - 7 t + 8 t - 3 t + t - 2 t + 1, [t]),6 5 4 3 2poly(- 2 t + t + t + t + t + t + 1, [t])],10 9 8 7 6 5 2[poly(t - 3 t + 3 t - 2 t + 2 t - t + t - 2 t + 1, [t]),7 4 3 2poly(- t + t + t + t + t + 1, [t])]}Now the set returned by the procedure consists of lists each containing two polynomials –the first polynomial is the numerator of the non-reduced Hilbert series, the second one thenumerator of the reduced Hilbert series (which had already been computed above).Since we can now compute all Hilbert series associated to a given Hilbert polynomial,we might be interested in computing all Hilbert functions associated to a specific Hilbertpolynomial. Thus, in the following section of the chapter we give an algorithm, whichcomputes the values of the Hilbert function from a given Hilbert series. In general, itsuffices to know the Hilbert series in order to compute the values of the Hilbert function –we will see that we can again benefit from our characterization of the lexicographic idealL p associated to some Hilbert polynomial p(z) to optimize our algorithm. Together withthe algorithm to compute all Hilbert series from a specific Hilbert polynomial discussedwithin this section, we obtain the possibility to compute all Hilbert functions to a givenHilbert polynomial.