Grobner bases of ideals of convergent power series

In extending Buchberger's theory[1.2] of Gröbner basis of polynomial ideals, Gröbner basis (standard basis in the notion of Hironaka[3]) of ideals containing power series has been discussed by several authors: Galligo[4] discussed reduction procedure of power series w.r.t. a given Gröbner basis, and Mora[5] derived a construction procedure of Gröbner basis in a local ring. In this paper, we formulate the Gröbner basis theory of convergent power series via truncated power series. In this formulation, finiteness and construction of Gröbner basis is proved quite simply. However, the termination of construction procedure remains an open problem although we have several results on this problem.