An efficient implementation for computing Gröbner bases over algebraic number fields

Abstract

In this paper we discuss Gröbner basis computation over algebraic number fields. Buchberger algorithm can be executed over any computable field, but the computation is often inefficient if the field operations for algebraic numbers are directly used. Instead we can execute the algorithm over the rationals by adding the defining polynomials to the input ideal and by setting an elimination order. In this paper we propose another method, which is a combination of the two methods above. We implement it in a computer algebra system Risa/Asir and examine its efficiency. © Springer-Verlag Berlin Heidelberg 2006.