488 research outputs found
Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting
Cylindrical algebraic decomposition (CAD) is an important tool, both for
quantifier elimination over the reals and a range of other applications.
Traditionally, a CAD is built through a process of projection and lifting to
move the problem within Euclidean spaces of changing dimension. Recently, an
alternative approach which first decomposes complex space using triangular
decomposition before refining to real space has been introduced and implemented
within the RegularChains Library of Maple. We here describe a freely available
package ProjectionCAD which utilises the routines within the RegularChains
Library to build CADs by projection and lifting. We detail how the projection
and lifting algorithms were modified to allow this, discuss the motivation and
survey the functionality of the package
An Incremental Algorithm for Computing Cylindrical Algebraic Decompositions
In this paper, we propose an incremental algorithm for computing cylindrical
algebraic decompositions. The algorithm consists of two parts: computing a
complex cylindrical tree and refining this complex tree into a cylindrical tree
in real space. The incrementality comes from the first part of the algorithm,
where a complex cylindrical tree is constructed by refining a previous complex
cylindrical tree with a polynomial constraint. We have implemented our
algorithm in Maple. The experimentation shows that the proposed algorithm
outperforms existing ones for many examples taken from the literature
Improved algorithm for computing separating linear forms for bivariate systems
We address the problem of computing a linear separating form of a system of
two bivariate polynomials with integer coefficients, that is a linear
combination of the variables that takes different values when evaluated at the
distinct solutions of the system. The computation of such linear forms is at
the core of most algorithms that solve algebraic systems by computing rational
parameterizations of the solutions and this is the bottleneck of these
algorithms in terms of worst-case bit complexity. We present for this problem a
new algorithm of worst-case bit complexity \sOB(d^7+d^6\tau) where and
denote respectively the maximum degree and bitsize of the input (and
where \sO refers to the complexity where polylogarithmic factors are omitted
and refers to the bit complexity). This algorithm simplifies and
decreases by a factor the worst-case bit complexity presented for this
problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields,
for this problem, a probabilistic Las-Vegas algorithm of expected bit
complexity \sOB(d^5+d^4\tau).Comment: ISSAC - 39th International Symposium on Symbolic and Algebraic
Computation (2014
The Slice Algorithm For Irreducible Decomposition of Monomial Ideals
Irreducible decomposition of monomial ideals has an increasing number of
applications from biology to pure math. This paper presents the Slice Algorithm
for computing irreducible decompositions, Alexander duals and socles of
monomial ideals. The paper includes experiments showing good performance in
practice.Comment: 25 pages, 8 figures. See http://www.broune.com/ for the data use
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