130 research outputs found
A survey on signature-based Gr\"obner basis computations
This paper is a survey on the area of signature-based Gr\"obner basis
algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain
the general ideas behind the usage of signatures. We show how to classify the
various known variants by 3 different orderings. For this we give translations
between different notations and show that besides notations many approaches are
just the same. Moreover, we give a general description of how the idea of
signatures is quite natural when performing the reduction process using linear
algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table
Signature Gr\"obner bases in free algebras over rings
We generalize signature Gr\"obner bases, previously studied in the free
algebra over a field or polynomial rings over a ring, to ideals in the mixed
algebra where is a principal
ideal domain. We give an algorithm for computing them, combining elements from
the theory of commutative and noncommutative (signature) Gr\"obner bases, and
prove its correctness.
Applications include extensions of the free algebra with commutative
variables, e.g., for homogenization purposes or for performing ideal theoretic
operations such as intersections, and computations over as
universal proofs over fields of arbitrary characteristic.
By extending the signature cover criterion to our setting, our algorithm also
lifts some technical restrictions from previous noncommutative signature-based
algorithms, now allowing, e.g., elimination orderings. We provide a prototype
implementation for the case when is a field, and show that our algorithm
for the mixed algebra is more efficient than classical approaches using
existing algorithms.Comment: 10 page
Axioms for a theory of signature bases
Twenty years after the discovery of the F5 algorithm, Gr\"obner bases with
signatures are still challenging to understand and to adapt to different
settings. This contrasts with Buchberger's algorithm, which we can bend in many
directions keeping correctness and termination obvious. I propose an axiomatic
approach to Gr\"obner bases with signatures with the purpose of uncoupling the
theory and the algorithms, and giving general results applicable in many
different settings (e.g. Gr\"obner for submodules, F4-style reduction,
noncommutative rings, non-Noetherian settings, etc.)
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