791 research outputs found
Simple and Nearly Optimal Polynomial Root-finding by Means of Root Radii Approximation
We propose a new simple but nearly optimal algorithm for the approximation of
all sufficiently well isolated complex roots and root clusters of a univariate
polynomial. Quite typically the known root-finders at first compute some crude
but reasonably good approximations to well-conditioned roots (that is, those
isolated from the other roots) and then refine the approximations very fast, by
using Boolean time which is nearly optimal, up to a polylogarithmic factor. By
combining and extending some old root-finding techniques, the geometry of the
complex plane, and randomized parametrization, we accelerate the initial stage
of obtaining crude to all well-conditioned simple and multiple roots as well as
isolated root clusters. Our algorithm performs this stage at a Boolean cost
dominated by the nearly optimal cost of subsequent refinement of these
approximations, which we can perform concurrently, with minimum processor
communication and synchronization. Our techniques are quite simple and
elementary; their power and application range may increase in their combination
with the known efficient root-finding methods.Comment: 12 pages, 1 figur
Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank
We show that there is a bound depending only on g and [K:Q] for the number of
K-rational points on a hyperelliptic curve C of genus g over a number field K
such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an
explicit bound is 8 r g + 33 (g - 1) + 1.
The proof is based on Chabauty's method; the new ingredient is an estimate
for the number of zeros of a logarithm in a p-adic `annulus' on the curve,
which generalizes the standard bound on disks. The key observation is that for
a p-adic field k, the set of k-points on C can be covered by a collection of
disks and annuli whose number is bounded in terms of g (and k).
We also show, strengthening a recent result by Poonen and the author, that
the lower density of hyperelliptic curves of odd degree over Q whose only
rational point is the point at infinity tends to 1 uniformly over families
defined by congruence conditions, as the genus g tends to infinity.Comment: 32 pages. v6: Some restructuring of the part of the argument relating
to annuli in hyperelliptic curves (some section numbers have changed),
various other improvements throughou
Connectedness properties of the set where the iterates of an entire function are bounded
We investigate some connectedness properties of the set of points K(f) where
the iterates of an entire function f are bounded. In particular, we describe a
class of transcendental entire functions for which an analogue of the
Branner-Hubbard conjecture holds and show that, for such functions, if K(f) is
disconnected then it has uncountably many components. We give examples to show
that K(f) can be totally disconnected, and we use quasiconformal surgery to
construct a function for which K(f) has a component with empty interior that is
not a singleton.Comment: 21 page
Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions
The parameter space for monic centered cubic polynomial maps
with a marked critical point of period is a smooth affine algebraic curve
whose genus increases rapidly with . Each consists of a
compact connectedness locus together with finitely many escape regions, each of
which is biholomorphic to a punctured disk and is characterized by an
essentially unique Puiseux series. This note will describe the topology of
, and of its smooth compactification, in terms of these escape
regions. It concludes with a discussion of the real sub-locus of
.Comment: 51 pages, 16 figure
Perturbations of quadratic centers of genus one
We propose a program for finding the cyclicity of period annuli of quadratic
systems with centers of genus one. As a first step, we classify all such
systems and determine the essential one-parameter quadratic perturbations which
produce the maximal number of limit cycles. We compute the associated
Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit
cycles. To illustrate our approach, we determine the cyclicity of the annuli of
two particular reversible systems.Comment: 28 page
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