957 research outputs found
Noncommutative Involutive Bases
The theory of Groebner Bases originated in the work of Buchberger and is now
considered to be one of the most important and useful areas of symbolic
computation. A great deal of effort has been put into improving Buchberger's
algorithm for computing a Groebner Basis, and indeed in finding alternative
methods of computing Groebner Bases. Two of these methods include the Groebner
Walk method and the computation of Involutive Bases. By the mid 1980's,
Buchberger's work had been generalised for noncommutative polynomial rings by
Bergman and Mora. This thesis provides the corresponding generalisation for
Involutive Bases and (to a lesser extent) the Groebner Walk, with the main
results being as follows. (1) Algorithms for several new noncommutative
involutive divisions are given, including strong; weak; global and local
divisions. (2) An algorithm for computing a noncommutative Involutive Basis is
given. When used with one of the aforementioned involutive divisions, it is
shown that this algorithm returns a noncommutative Groebner Basis on
termination. (3) An algorithm for a noncommutative Groebner Walk is given, in
the case of conversion between two harmonious monomial orderings. It is shown
that this algorithm generalises to give an algorithm for performing a
noncommutative Involutive Walk, again in the case of conversion between two
harmonious monomial orderings. (4) Two new properties of commutative involutive
divisions are introduced (stability and extendibility), respectively ensuring
the termination of the Involutive Basis algorithm and the applicability (under
certain conditions) of homogeneous methods of computing Involutive Bases.Comment: 378+x+I Pages; PhD Thesis (University of Wales, Bangor); Code
available at http://www.dilan4.freeserve.co.uk/maths
A new algorithm for computing Groebner bases
Buchberger\u27s algorithm for computing Groebner bases was introduced in 1965, and subsequently there have been extensive efforts in improving its efficiency. Major algorithms include F4 (Faugère 1999), XL (Courtois et al. 2000) and F5 (Faugère 2002). F5 is believed to be the fastest algorithm known in the literature. Most recently, Gao, Guan and Volny (2010) introduced an incremental algorithm (G2V) that is simpler and several times faster than F5. In this paper, a new algorithm is presented that can avoid the incremental nature of F5 and G2V. It matches Buchberger\u27s algorithm in simplicity and yet is more flexible. More precisely, 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 given list of polynomials. For any term order for the ideal, one may vary signature orders (i.e. the term orders for the syzygy module). Under one signature order, the new algorithm specializes to the G2V, and under another signature order, the new algorithm is several times faster than G2V, as indicated by computer experiments on benchmark examples
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)
An Algebraic Model For Quorum Systems
Quorum systems are a key mathematical abstraction in distributed
fault-tolerant computing for capturing trust assumptions. A quorum system is a
collection of subsets of all processes, called quorums, with the property that
each pair of quorums have a non-empty intersection. They can be found at the
core of many reliable distributed systems, such as cloud computing platforms,
distributed storage systems and blockchains. In this paper we give a new
interpretation of quorum systems, starting with classical majority-based quorum
systems and extending this to Byzantine quorum systems. We propose an algebraic
representation of the theory underlying quorum systems making use of
multivariate polynomial ideals, incorporating properties of these systems, and
studying their algebraic varieties. To achieve this goal we will exploit
properties of Boolean Groebner bases. The nice nature of Boolean Groebner bases
allows us to avoid part of the combinatorial computations required to check
consistency and availability of quorum systems. Our results provide a novel
approach to test quorum systems properties from both algebraic and algorithmic
perspectives.Comment: 15 pages, 3 algorithm
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
Parallel and distributed Gr\"obner bases computation in JAS
This paper considers parallel Gr\"obner bases algorithms on distributed
memory parallel computers with multi-core compute nodes. We summarize three
different Gr\"obner bases implementations: shared memory parallel, pure
distributed memory parallel and distributed memory combined with shared memory
parallelism. The last algorithm, called distributed hybrid, uses only one
control communication channel between the master node and the worker nodes and
keeps polynomials in shared memory on a node. The polynomials are transported
asynchronous to the control-flow of the algorithm in a separate distributed
data structure. The implementation is generic and works for all implemented
(exact) fields. We present new performance measurements and discuss the
performance of the algorithms.Comment: 14 pages, 8 tables, 13 figure
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