53,892 research outputs found
On the asymptotic and practical complexity of solving bivariate systems over the reals
This paper is concerned with exact real solving of well-constrained,
bivariate polynomial systems. The main problem is to isolate all common real
roots in rational rectangles, and to determine their intersection
multiplicities. We present three algorithms and analyze their asymptotic bit
complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based
method, and \sOB(N^{12}) for two subresultant-based methods: this notation
ignores polylogarithmic factors, where bounds the degree and the bitsize of
the polynomials. The previous record bound was \sOB(N^{14}).
Our main tool is signed subresultant sequences. We exploit recent advances on
the complexity of univariate root isolation, and extend them to sign evaluation
of bivariate polynomials over two algebraic numbers, and real root counting for
polynomials over an extension field. Our algorithms apply to the problem of
simultaneous inequalities; they also compute the topology of real plane
algebraic curves in \sOB(N^{12}), whereas the previous bound was
\sOB(N^{14}).
All algorithms have been implemented in MAPLE, in conjunction with numeric
filtering. We compare them against FGB/RS, system solvers from SYNAPS, and
MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is
among the most robust, and its runtimes are comparable, or within a small
constant factor, with respect to the C/C++ libraries.
Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure
On the Skolem Problem for Continuous Linear Dynamical Systems
The Continuous Skolem Problem asks whether a real-valued function satisfying
a linear differential equation has a zero in a given interval of real numbers.
This is a fundamental reachability problem for continuous linear dynamical
systems, such as linear hybrid automata and continuous-time Markov chains.
Decidability of the problem is currently open---indeed decidability is open
even for the sub-problem in which a zero is sought in a bounded interval. In
this paper we show decidability of the bounded problem subject to Schanuel's
Conjecture, a unifying conjecture in transcendental number theory. We
furthermore analyse the unbounded problem in terms of the frequencies of the
differential equation, that is, the imaginary parts of the characteristic
roots. We show that the unbounded problem can be reduced to the bounded problem
if there is at most one rationally linearly independent frequency, or if there
are two rationally linearly independent frequencies and all characteristic
roots are simple. We complete the picture by showing that decidability of the
unbounded problem in the case of two (or more) rationally linearly independent
frequencies would entail a major new effectiveness result in Diophantine
approximation, namely computability of the Diophantine-approximation types of
all real algebraic numbers.Comment: Full version of paper at ICALP'1
Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces
This paper transfers a randomized algorithm, originally used in geometric
optimization, to computational problems in commutative algebra. We show that
Clarkson's sampling algorithm can be applied to two problems in computational
algebra: solving large-scale polynomial systems and finding small generating
sets of graded ideals. The cornerstone of our work is showing that the theory
of violator spaces of G\"artner et al.\ applies to polynomial ideal problems.
To show this, one utilizes a Helly-type result for algebraic varieties. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the ideas interesting for handling systems with very large
numbers of polynomials, but whose rank in the vector space of polynomials is
small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange
Some Speed-Ups and Speed Limits for Real Algebraic Geometry
We give new positive and negative results (some conditional) on speeding up
computational algebraic geometry over the reals: (1) A new and sharper upper
bound on the number of connected components of a semialgebraic set. Our bound
is novel in that it is stated in terms of the volumes of certain polytopes and,
for a large class of inputs, beats the best previous bounds by a factor
exponential in the number of variables. (2) A new algorithm for approximating
the real roots of certain sparse polynomial systems. Two features of our
algorithm are (a) arithmetic complexity polylogarithmic in the degree of the
underlying complex variety (as opposed to the super-linear dependence in
earlier algorithms) and (b) a simple and efficient generalization to certain
univariate exponential sums. (3) Detecting whether a real algebraic surface
(given as the common zero set of some input straight-line programs) is not
smooth can be done in polynomial time within the classical Turing model (resp.
BSS model over C) only if P=NP (resp. NP<=BPP). The last result follows easily
from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of
Complexity. More typos are corrected, and a new section is added where the
bounds here are compared to an earlier result of Benedetti, Loeser, and
Risler. The LaTeX source needs the ajour.cls macro file to compil
Computing Equilibria of Semi-algebraic Economies Using Triangular Decomposition and Real Solution Classification
In this paper, we are concerned with the problem of determining the existence
of multiple equilibria in economic models. We propose a general and complete
approach for identifying multiplicities of equilibria in semi-algebraic
economies, which may be expressed as semi-algebraic systems. The approach is
based on triangular decomposition and real solution classification, two
powerful tools of algebraic computation. Its effectiveness is illustrated by
two examples of application.Comment: 24 pages, 5 figure
Heights and totally real numbers
1973 Schinzel proved that the standard logarithmic height h on the maximal
totally real field extension of the rationals is either zero or bounded from
below by a positive constant. In this paper we study this property for
canonical heights associated to rational functions and the corresponding
dynamical system on the affine line. At the end, we will give a few remarks on
the behavior of h on finite extensions of the maximal totally real field.Comment: Major changes regarding the first version. E.g. the last chapter was
cancele
Report on some recent advances in Diophantine approximation
A basic question of Diophantine approximation, which is the first issue we
discuss, is to investigate the rational approximations to a single real number.
Next, we consider the algebraic or polynomial approximations to a single
complex number, as well as the simultaneous approximation of powers of a real
number by rational numbers with the same denominator. Finally we study
generalisations of these questions to higher dimensions. Several recent
advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent,
T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these
works.Comment: to be published by Springer Verlag, Special volume in honor of Serge
Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakrishnan, Ken Ribet and
John Tat
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