Gröbner bases over algebraic number fields

Abstract

Abstract. Although Buchberger's algorithm, in theory, allows us to compute Gröbner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field K = Q(α), a simple extension of Q, where α is an algebraic number, and let f ∈ Q[t] be the minimal polynomial of α. In this paper we present a new efficient method to compute Gröbner bases in polynomial rings over the algebraic number field K. Starting from the ideas of Noro [11], we proceed by joining f to the ideal to be considered, adding t as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2, 3,10], that is, by inferring information in characteristic zero from information in characteristic p > 0. For suitable primes p, the minimal polynomial f is reducible over Fp. This allows us to apply modular methods once again, on a second level, with respect to the factors of f . The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. At current state, the algorithm is probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithm, which has been implemented in Singular [7], outperforms other known methods by far.

Current Challenges in Developing Open Source Computer Algebra Systems

“…One general way of achieving higher speed is the parallelization of algorithms. This will be addressed in the next section, where we will, in particular, discuss a parallel version of the Gröbner basis (syzygy) algorithm which is specific to number fields [18]. New ideas for enhancing syzygy computations in general are presented in [31].…”

“…For the fundamental task of computing Gröbner bases, a modular version of Buchberger's algorithm is due to Arnold [1]. More recently, Boku, Fieker, Steenpaß and the second author [18] have designed a modular Gröbner bases algorithm which is specific to number fields. In addition to using the approach from Arnold's paper, which is to compute Gröbner bases modulo several primes and then use Chinese remaindering together with rational reconstruction, the new approach provides a second level of parallelization as depicted in Figure 3: If the number field is presented as…”