Computing Left Gr（o|¨）bner Bases Over The Solvable Polynomial Algebra K（（?））[a₁,…,a_n]

Let θbe an algebraic element over the field K,p(x)∈K[x] the minimal polynomial of θ over K,and A=K(θ)[a1,…,an]a solvable polynomial algebra of n generators over the simple algebraic extension K(θ).It is shown that if the subalgebra K[al,…,an] of A is a solvable polynomial algebra over K then A=K[a1,…,an][x],the polynomial algebra in one commuting variable x over k[a1,…,an],is a solvable polynomial algebra generated by n+1elements over K；that a left Grobner basis of any left ideal generated by m elements{f1,…,.fm)in A can be obtained by computing a left Grobner basis of the left ideal generated by m+1elements {f1,…,fm,p(x))in A;and that if L=(?)ri=1Aei is the free A-module of rank r andL=(?)=1Aei is the free A-module of rank r,then a left Grobner basis for a submodule of L generated by s elements{ξ1….,ξs)can be obtained by computing a left Grobner basis of a submodule of Lgenerated by s+r elements{ξ1,…,ξs,p(x)el,…,p(x)er}.Based on the proof of the main results,the detailed procedures of transferring the computation of left Grobner bases over K(θ) to the computation over K are clearly given,and the feasibility of the proposed method is examined by using the computer algebra system Singular3-1-4.