University of Paderborn Department of Mathematics Diploma Thesis ...

4.4. COMPUTING EXPANSIONS AND CONTRACTIONS 89Ind:= Ind union {j};end_if;end_for;end_for;v:= 0;for ind in Ind doM[ind-v]:= null();v:= v+1;end_for;/* Delete m from M and insert the set of newminimal generators into M */M:= [ M[j] $ j = 1..contains(M,m)-1,op(newMonomials),M[j] $ j = contains(M,m)+1..nops(M) ];return(M);end_proc:Although the above lines might look a little complicated, this is not really the case. Thecomputation of the set of minimal generators after performing the expansion of somemonomial m uses nothing more than the original definition of the expansion. The only fact,which makes the source code look more complicated than perhaps expected is that we haveto check if all monomials, which are minimal generators in the set M before the expansion,are still minimal generators after expanding m. To guarantee that the procedure alwaysreturns a set of minimal generators again, it suffices to consider all monomials m ′ ∈ M tothe right of m. Hence, we check for all of these monomials m ′ in M, if they are divisibleby any of the monomials, which are included into M by the expansion of m. If this is thecase, we can remove such elements m ′ from M. Some examples will illustrate the use ofthis new procedure.Example 4.10. We consider the ideals I 1 := (x 2 0, x 0 x 1 , x 0 x 2 , x 3 1) ⊂ K[x 0 , x 1 , x 2 , x 3 ] andI 2 := (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 6 3) ⊂ K[x 0 , x 1 , x 2 , x 3 , x 4 ].(i) First we expand the monomial x 0 x 2 in I 1 :MuPAD>> compute_expansion([[2,0,0,0],[1,1,0,0],[1,0,1,0],[0,3,0,0]],>> [1,0,1,0], 3);Output[[2, 0, 0, 0], [1, 1, 0, 0], [1, 0, 2, 0], [0, 3, 0, 0]]