15,163 research outputs found
New Algorithms for Computing Groebner Bases
In this thesis, we present new algorithms for computing Groebner bases. The first algorithm, G2V, is incremental in the same fashion as F5 and F5C. At a typical step, one is given a Groebner basis G for an ideal I and any polynomial g, and it is desired to compute a Groebner basis for the new ideal , obtained from I by joining g. Let (I : g) denote the colon ideal of I divided by g. Our algorithm computes Groebner bases for I, g and (I : g) simultaneously. In previous algorithms, S-polynomials that reduce to zero are useless, in fact, F5 tries to avoid such reductions as much as possible. In our algorithm, however, these \u27useless\u27 S-polynomials give elements in (I : g) and are useful in speeding up the subsequent computations. Computer experiments on some benchmark examples indicate that our algorithm is much more efficient (two to ten times faster) than F5 and F5C. Next, we present a more general algorithm that matches Buchberger\u27s algorithm in simplicity and yet is more flexible than G2V. Given a list of polynomials, the new algorithm computes simultaneously a Groebner basis for the ideal generated by the polynomials and a Groebner basis for the leading terms of the syzygy module of the polynomials. For any term order for the ideal, one may vary the term order for the syzygy module. Under one term order for the syzygy module, the new algorithm specializes to the G2V algorithm, and under another term order for the syzygy module, the new algorithm may be several times faster than G2V, as indicated by computer experiments on benchmark examples. Finally, we present a solid theoretical framework for G2V and GVW which makes the algorithm much more understandable. This theory also gives a major improvement of the GVW algorithm. A proof of termination is provided for all algorithms, and an argument is made that GVW computes the fewest number of generators for the signature based algorithms used by GVW and F5 (similarly for G2V and F5C)
Modifying Faug\`ere's F5 Algorithm to ensure termination
The structure of the F5 algorithm to compute Gr\"obner bases makes it very
efficient. However, while it is believed to terminate for so-called regular
sequences, it is not clear whether it terminates for all inputs. This paper has
two major parts. In the first part, we describe in detail the difficulties
related to a proof of termination. In the second part, we explore three
variants that ensure termination. Two of these have appeared previously only in
dissertations, and ensure termination by checking for a Gr\"obner basis using
traditional criteria. The third variant, F5+, identifies a degree bound using a
distinction between "necessary" and "redundant" critical pairs that follows
from the analysis in the first part. Experimental evidence suggests this third
approach is the most efficient of the three.Comment: 19 pages, 1 tabl
Termination of Original F5
The original F5 algorithm introduced by Faug\`ere is formulated for any
homogeneous polynomial set input. The correctness of output is shown for any
input that terminates the algorithm, but the termination itself is proved only
for the case of input being regular polynomial sequence. This article shows
that algorithm correctly terminates for any homogeneous input without any
reference to regularity. The scheme contains two steps: first it is shown that
if the algorithm does not terminate it eventually generates two polynomials
where first is a reductor for the second. But first step does not show that
this reduction is permitted by criteria introduced in F5. The second step shows
that if such pair exists then there exists another pair for which the reduction
is permitted by all criteria. Existence of such pair leads to contradiction.
Version v3 fixes the bibliography
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
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
Gr\"obner Bases of Bihomogeneous Ideals generated by Polynomials of Bidegree (1,1): Algorithms and Complexity
Solving multihomogeneous systems, as a wide range of structured algebraic
systems occurring frequently in practical problems, is of first importance.
Experimentally, solving these systems with Gr\"obner bases algorithms seems to
be easier than solving homogeneous systems of the same degree. Nevertheless,
the reasons of this behaviour are not clear. In this paper, we focus on
bilinear systems (i.e. bihomogeneous systems where all equations have bidegree
(1,1)). Our goal is to provide a theoretical explanation of the aforementionned
experimental behaviour and to propose new techniques to speed up the Gr\"obner
basis computations by using the multihomogeneous structure of those systems.
The contributions are theoretical and practical. First, we adapt the classical
F5 criterion to avoid reductions to zero which occur when the input is a set of
bilinear polynomials. We also prove an explicit form of the Hilbert series of
bihomogeneous ideals generated by generic bilinear polynomials and give a new
upper bound on the degree of regularity of generic affine bilinear systems.
This leads to new complexity bounds for solving bilinear systems. We propose
also a variant of the F5 Algorithm dedicated to multihomogeneous systems which
exploits a structural property of the Macaulay matrix which occurs on such
inputs. Experimental results show that this variant requires less time and
memory than the classical homogeneous F5 Algorithm.Comment: 31 page
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