9,467 research outputs found
Continued Fraction Expansion of Real Roots of Polynomial Systems
We present a new algorithm for isolating the real roots of a system of
multivariate polynomials, given in the monomial basis. It is inspired by
existing subdivision methods in the Bernstein basis; it can be seen as
generalization of the univariate continued fraction algorithm or alternatively
as a fully analog of Bernstein subdivision in the monomial basis. The
representation of the subdivided domains is done through homographies, which
allows us to use only integer arithmetic and to treat efficiently unbounded
regions. We use univariate bounding functions, projection and preconditionning
techniques to reduce the domain of search. The resulting boxes have optimized
rational coordinates, corresponding to the first terms of the continued
fraction expansion of the real roots. An extension of Vincent's theorem to
multivariate polynomials is proved and used for the termination of the
algorithm. New complexity bounds are provided for a simplified version of the
algorithm. Examples computed with a preliminary C++ implementation illustrate
the approach.Comment: 10 page
On Continued Fraction Expansion of Real Roots of Polynomial Systems, Complexity and Condition Numbers
International audienceWe elaborate on a correspondence between the coeffcients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions, that use only integer arithmetic (in contrast to Bernstein basis) and are feasible over unbounded regions. Then, we study an algorithm to split this representation and we obtain a subdivision scheme for the domain of multivariate polynomial functions. This implies a new algorithm for real root isolation, MCF, that generalizes the Continued Fraction (CF) algorithm of univariate polynomials. A partial extension of Vincent's Theorem for multivariate polynomials is presented, which allows us to prove the termination of the algorithm. Bounding functions, projection and preconditioning are employed to speed up the scheme. The resulting isolation boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. Finally, we present new complexity bounds for a simplified version of the algorithm in the bit complexity model, and also bounds in the real RAM model for a family of subdivision algorithms in terms of the real condition number of the system. Examples computed with our C++ implementation illustrate the practical aspects of our method
The first rational Chebyshev knots
A Chebyshev knot is a knot which has a parametrization
of the form where
are integers, is the Chebyshev polynomial of degree and We show that any two-bridge knot is a Chebyshev knot with and also
with . For every integers ( and , coprime), we
describe an algorithm that gives all Chebyshev knots \cC(a,b,c,\phi). We
deduce a list of minimal Chebyshev representations of two-bridge knots with
small crossing number.Comment: 22p, 27 figures, 3 table
Structured matrices, continued fractions, and root localization of polynomials
We give a detailed account of various connections between several classes of
objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices,
Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems,
total positivity, and root localization of univariate polynomials. Along with a
survey of many classical facts, we provide a number of new results.Comment: 79 pages; new material added to the Introductio
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
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