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

摘要

Let A = k[a(1),..., a(n)] be an affine algebra over a field k of characteristic 0, and let FA = (F(n)A}(n)greater than or equal to(0) be the standard filtration on A. Consider the graded algebras associated with A: G(A) = circle plus(p)greater than or equal to(0)(F(p)A/Fp-1 A), the associated graded algebra of A, and (A) over tilde = circle plus(p greater than or equal to 0)F(p)A, the Rees algebra of A. It is proved that A is a solvable polynomial algebra with >(grlex) in the sense of [K-RW] if and only if G(A) is a solvable polynomial algebra with >(grlex) if and only if (A) over tilde is a solvable polynomial algebra with >(grlex). Suppose that A is a solvable polynomial algebra with >(grlex). Let L be a left ideal of A and F = (f(1),..., f(s)} subset of L. It is proved that F is a (left) Groebner basis for L in the sense of [K-RW] if and only if sigma(F) = {sigma(f(1)),..., sigma(f(s))} is a (left) Groebner basis for G(L) in G(A) if and only if (F) over tilde = {(f) over tilde 1,..., (f) over tilde s} is a (left) Groebner basis for (L) over tilde, where G(L) resp. (L) over tilde is the associated graded left ideal of L in G(A) resp. the associated graded left ideal of L in A with respect to the filtration FL induced by FA on L, and the sigma(f(i))'s resp. (f) over tilde(i)'s are corresponding homogeneous elements of the f(i)'s in G(L) resp. in (L) over tilde. Examples of applications of this filtered-graded transfer method to some popular algebras are given. [References: 23]