Solving Parametric Ideal Membership Problems and Computing Integral Numbers in a Ring of Convergent Power Series Via Comprehensive Gröbner Systems

Abstract

New algorithms are presented for solving ideal membership problems with parameters and for computing integral numbers in a ring of convergent power series. It is shown that the ideal membership problems for zero-dimensional ideals in the ring of convergent power series, can be solved in the polynomial ring. The key idea of the algorithms is computing comprehensive Grobner systems of ideal quotients in a polynomial ring. Furthermore, new methods for computing integral numbers, in the local ring, are introduced as an application.