University of Paderborn Department of Mathematics Diploma Thesis ...

4.4. COMPUTING EXPANSIONS AND CONTRACTIONS 87more variable included into the set of generators of the unique lexicographic ideal L p if weincrease the value of n by 1:(iv) p(z) = 2z 2 + 2z + 1 and R := K[x 0 , x 1 , x 2 , x 3 , x 5 ], i.e. n = 5.MuPAD>> compute_L_p(poly(2*z^2 + 2*z + 1,[z]), 5);Output[[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0],[0, 0, 4, 3, 0, 0], [0, 0, 4, 2, 6, 0]]i.e. L p = (x 0 , x 1 , x 5 2, x 4 2x 3 3, x 4 2x 2 3x 6 4).4.4 Computing expansions and contractionsIn the last section of Chapter 3, we frequently used expansions and contractions of monomialsas one of the central tools to prove that all saturated stable ideals with the same Hilbertpolynomial (and the same double saturation) are linked to each other by expansions andcontractions. Furthermore, we saw in Corollary 3.37 that to a given Hilbert polynomialp(z), we can compute all unique lexicographic ideals L H to Hilbert series H(t) associatedto the Hilbert polynomial p(z) from the unique lexicographic ideal L p by expansions andcontractions. Within this section we present the MuPAD source code to compute expansionsand contractions with the computer algebra system.Again we will use lists to encode monomials and a list of lists to encode the set of generatorsof any stable ideal.MuPAD Source Code 4.9. The procedure below presents a possible implementation ofthe expansion of a monomial m ∈ M in a stable ideal I ⊂ R. The input for the procedureis defined to be the set of minimal generators M of a stable ideal I in descending lexicographicorder, a monomial m, which is expandable in I, and the index n of the last variablein R. It returns a list of lists encoding the new set of minimal generators in lexicographicorder.Input for the procedure compute expansion.◦ M — a list of lists encoding the set of generators of some stable ideal in lexicographicorder◦ m — a list encoding the monomial to be expanded◦ n — the index of the last variable of the polynomial ring R