Solving Extended Ideal Membership Problems in Rings of Convergent Power Series via Gröbner Bases

Abstract

An extended ideal membership algorithm is considered in the ring of convergent power series. It is shown that the problem for zero-dimensional ideals in a local ring can be solved in a polynomial ring. The key of the proposed method is the use of ideal quotients in polynomial rings. A new algorithm is given to solve the extended ideal membership problems in local rings. A generalization of the resulting algorithm to ideals with parameters is also described.