Filtered-graded transfer of groebner basis computation in solvable polynomial algebras

Abstract

Let be an affine algebra over a field k of characteristic 0, and let be the standard filtration on A. Consider the graded algebras associated with , the associated graded algebra of A, and , the Rees algebra of A. It is proved that A is a solvable polynomial algebra with in the sense of [K-RW] if and only if G(A) is a solvable polynomial algebra with if and only if is a solvable polynomial algebra with Suppose that A is a solvable polynomial algebra with , Let L be a left ideal of A and . It is proved that F is a (left) Groebner basis for L in the sense of [K-RW] if and only if , is a (left) Groebner basis for G(L) in G(A) if and only if is a (left) Groebner basis for , where G(L) resp. is the associated graded left ideal of L in G(A) resp. the associated graded left ideal of L in with respect to the filtration FL induced by FA on L, and the resp. are corresponding homogeneous elements of the in G(L) resp. in . Examples of applications of this filtered-graded transfer method to some popular algebras are given.