132 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
Syzygies among reduction operators
We introduce the notion of syzygy for a set of reduction operators and relate
it to the notion of syzygy for presentations of algebras. We give a method for
constructing a linear basis of the space of syzygies for a set of reduction
operators. We interpret these syzygies in terms of the confluence property from
rewriting theory. This enables us to optimise the completion procedure for
reduction operators based on a criterion for detecting useless reductions. We
illustrate this criterion with an example of construction of commutative
Gr{\"o}bner basis
The F5 Criterion revised
The purpose of this work is to generalize part of the theory behind Faugere's
"F5" algorithm. This is one of the fastest known algorithms to compute a
Groebner basis of a polynomial ideal I generated by polynomials
f_{1},...,f_{m}. A major reason for this is what Faugere called the algorithm's
"new" criterion, and we call "the F5 criterion"; it provides a sufficient
condition for a set of polynomials G to be a Groebner basis. However, the F5
algorithm is difficult to grasp, and there are unresolved questions regarding
its termination.
This paper introduces some new concepts that place the criterion in a more
general setting: S-Groebner bases and primitive S-irreducible polynomials. We
use these to propose a new, simple algorithm based on a revised F5 criterion.
The new concepts also enable us to remove various restrictions, such as proving
termination without the requirement that f_{1},...,f_{m} be a regular sequence.Comment: Originally submitted by Arri in 2009, with material added by Perry
since 2010. The 2016 editions correct typographical issues not caught in
previous editions bring the theory of the body into conformity with the
published version of the pape
Computing periods of rational integrals
A period of a rational integral is the result of integrating, with respect to
one or several variables, a rational function over a closed path. This work
focuses particularly on periods depending on a parameter: in this case the
period under consideration satisfies a linear differential equation, the
Picard-Fuchs equation. I give a reduction algorithm that extends the
Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs
equations. The resulting algorithm is elementary and has been successfully
applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at
http://pierre.lairez.fr/supp/periods
A Tropical F5 algorithm
Let K be a field equipped with a valuation. Tropical varieties over K can be
defined with a theory of Gr{\"o}bner bases taking into account the valuation of
K. While generalizing the classical theory of Gr{\"o}bner bases, it is not
clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to
the tropical case. Among them, one of the most efficient is the celebrated F5
Algorithm of Faug{\`e}re. In this article, we prove that, for homogeneous
ideals, it can be adapted to the tropical case. We prove termination and
correctness. Because of the use of the valuation, the theory of tropical
Gr{\"o}b-ner bases is promising for stable computations over polynomial rings
over a p-adic field. We provide numerical examples to illustrate
time-complexity and p-adic stability of this tropical F5 algorithm
An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
In this paper we develop an axiomatic setup for algorithmic homological
algebra of Abelian categories. This is done by exhibiting all existential
quantifiers entering the definition of an Abelian category, which for the sake
of computability need to be turned into constructive ones. We do this
explicitly for the often-studied example Abelian category of finitely presented
modules over a so-called computable ring , i.e., a ring with an explicit
algorithm to solve one-sided (in)homogeneous linear systems over . For a
finitely generated maximal ideal in a commutative ring we
show how solving (in)homogeneous linear systems over can be
reduced to solving associated systems over . Hence, the computability of
implies that of . As a corollary we obtain the computability
of the category of finitely presented -modules as an Abelian
category, without the need of a Mora-like algorithm. The reduction also yields,
as a by-product, a complexity estimation for the ideal membership problem over
local polynomial rings. Finally, in the case of localized polynomial rings we
demonstrate the computational advantage of our homologically motivated
alternative approach in comparison to an existing implementation of Mora's
algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu
Toric Border Bases
We extend the theory and the algorithms of Border Bases to systems of Laurent
polynomial equations, defining "toric" roots. Instead of introducing new
variables and new relations to saturate by the variable inverses, we propose a
more efficient approach which works directly with the variables and their
inverse. We show that the commutation relations and the inversion relations
characterize toric border bases. We explicitly describe the first syzygy module
associated to a toric border basis in terms of these relations. Finally, a new
border basis algorithm for Laurent polynomials is described and a proof of its
termination is given for zero-dimensional toric ideals
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
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