Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gr\"obner Bases over Algebraic Number Fields   Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner
bases over any field, in practice, however, the computational efficiency
depends on the arithmetic of the ground field. Consider a field $K =
\mathbb{Q}(\alpha)$, a simple extension of $\mathbb{Q}$, where $\alpha$ is an
algebraic number, and let $f \in \mathbb{Q}[t]$ be the minimal polynomial of
$\alpha$. In this paper we present a new efficient method to compute Gr\"obner
bases in polynomial rings over the algebraic number field $K$. Starting from
the ideas of Noro [Noro, 2006], 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 [Arnold, 2003; B\"ohm et al., 2015;
Idrees et al., 2011], 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 $\mathbb{F}_p$. 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 Gr\"obner 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,
outperforms other known methods by far.

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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
3
The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an