Computing Over K And Z The Gro(o|¨)bner Bases With Coefficients In K(α) And Z[α]

Let a be an algebraic element over a number field K, p(t)∈K[t] the minimal polynomial of a over K, A=K(α)[x1,..., xn] the polynomial algebra in n variables over the simple algebraic extension K(a) of K, L=(?)im=1Aei the free A-module of rank m; and let A=K[t,x1,..., xn] be the polynomial algebra in n+1variables over K, L=(?)im=1Aei the free A-module of rank m. It is proved that a Grobner basis of an A-submodule of L generated by q elements{ξ1,...,ξq} can be obtained by computing a Grobner basis of the A-submodule of L generated by q+m elements {ξ1,……ξq,p(t)e1,……,P(t)em};at the same time, the procedure of transferring the computation is clearly given. The high efficiency of the proposed method is examined by using the computer algebra system Maple-14.Moreover, let α∈C be an algebraic integer over the ring of integers Z, and Z[α] the simple algebraic extension ring of Z; Also, let A=Z[a][x1,...,xn) be the polynomial ring in n variables over Z[a], and A=Z[t,x1,,...,xn] the polynomial ring in n+1variables over Z. Then, in the way similar to the discussion over a field, it is proved that a Grobner basis of an ideal I of A can be obtained by computing a Grobner basis of some ideal I in A; at the same time, the procedure of transferring the computation is clearly given. The proposed method is illustrated by an example using the computer algebra system Macaulay2.