Modular algorithms for computing Gröbner bases

Arnold, Elizabeth

doi:10.1016/s0747-7171(02)00140-2

Cited by 104 publications

(95 citation statements)

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“…In the second step, following [2,10], we use the Chinese remainder algorithm for integers together with rational reconstruction to lift these results to the reduced Gröbner basis G of I. In the last step, we lift G to a Gröbner basis G of I over K by mapping t to α (see Theorem 5.1).…”

Section: Instead Of Computing In the Ringmentioning

confidence: 99%

“…One of the reasons for this is that over the field of rational numbers, we often suffer from coefficient swell. Various methods to avoid this have been investigated; the trace algorithm [13] and modular algorithms [2,10] are successful in this direction. But using these approaches, we still have to deal with the complicated arithmetic in algebraic number fields, in particular with the computation of inverses.…”

Section: Introductionmentioning

confidence: 99%

“…Starting from a polynomial ring over Q as explained above, we apply the modular methods for computing Gröbner bases discussed in [2,3,10] to pass to positive characteristic p. Choosing a set P of suitable prime numbers, see Definition 5.2, the image f p of f in F p [t] is, for p ∈ P, reducible and square-free. We can thus again apply modular methods, with respect to the factors f 1,p , .…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…K = ( 1 , ) 4: lift G to G via the map sending t to α 5: return G Algorithm 2 is probabilistic in the sense that the test in lines 17 to 18 does not guarantee that G = I. If I is homogeneous, however, the result G of Algorithm 3 can be verified along the lines of [2,Theorem 7.1]. With this test included, Algorithm 3 is deterministic.…”

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confidence: 99%

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Gröbner bases over algebraic number fields

Boku

Decker

Fieker

et al. 2015

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation

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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.

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Section: Instead Of Computing In the Ringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%