University of Paderborn Department of Mathematics Diploma Thesis ...

92 CHAPTER 4. ALGORITHMS FOR STABLE IDEALSif m[i] - M[j][i] > 0 thendel:= del union {1};elif m[i] - M[j][i] < 0 thendel:= del union {-1};end_if;end_for;if contains(del, 1) = FALSE orcontains(del, -1) = FALSE thenInd:= Ind union {j};end_if;end_for;v:= 0;for ind in Ind doM[ind-v]:= null();v:= v+1;end_for;return(M);end_proc:The above procedure only uses the original definition of the contraction of a monomial.Again, as in 4.9, we wish to guarantee that the set of monomials returned by the algorithmis again a set of minimal generators. Hence, we first include m into the set M (note thatsince we contract m, m itself is not an element of the input set M) and then check if anymonomial m ′ to the left of m is divisible by m. If this is the case, we can remove m ′ fromM.Again, we consider a few examples:Example 4.12. Let R := K[x 0 , x 1 , x 2 , x 3 , x 4 ], i.e. n = 4.(i) Consider the ideal I 1 := (x 2 0, x 0 x 1 , x 0 x 2 , x 0 x 3 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 6 3). Contraction of x 0 ispossible and providesMuPAD>> compute_contraction([[2,0,0,0,0],[1,1,0,0,0],[1,0,1,0,0],>> [1,0,0,1,0],[0,5,0,0,0],[0,4,3,0,0],>> [0,4,2,6,0]],>> [1,0,0,0,0], 4);Output[[1, 0, 0, 0, 0], [0, 5, 0, 0, 0], [0, 4, 3, 0, 0], [0, 4, 2, 6, 0]]