1,298 research outputs found
Computing Bits of Algebraic Numbers
We initiate the complexity theoretic study of the problem of computing the
bits of (real) algebraic numbers. This extends the work of Yap on computing the
bits of transcendental numbers like \pi, in Logspace.
Our main result is that computing a bit of a fixed real algebraic number is
in C=NC1\subseteq Logspace when the bit position has a verbose (unary)
representation and in the counting hierarchy when it has a succinct (binary)
representation.
Our tools are drawn from elementary analysis and numerical analysis, and
include the Newton-Raphson method. The proof of our main result is entirely
elementary, preferring to use the elementary Liouville's theorem over the much
deeper Roth's theorem for algebraic numbers.
We leave the possibility of proving non-trivial lower bounds for the problem
of computing the bits of an algebraic number given the bit position in binary,
as our main open question. In this direction we show very limited progress by
proving a lower bound for rationals
Computing low-degree factors of lacunary polynomials: a Newton-Puiseux approach
We present a new algorithm for the computation of the irreducible factors of
degree at most , with multiplicity, of multivariate lacunary polynomials
over fields of characteristic zero. The algorithm reduces this computation to
the computation of irreducible factors of degree at most of univariate
lacunary polynomials and to the factorization of low-degree multivariate
polynomials. The reduction runs in time polynomial in the size of the input
polynomial and in . As a result, we obtain a new polynomial-time algorithm
for the computation of low-degree factors, with multiplicity, of multivariate
lacunary polynomials over number fields, but our method also gives partial
results for other fields, such as the fields of -adic numbers or for
absolute or approximate factorization for instance.
The core of our reduction uses the Newton polygon of the input polynomial,
and its validity is based on the Newton-Puiseux expansion of roots of bivariate
polynomials. In particular, we bound the valuation of where is
a lacunary polynomial and a Puiseux series whose vanishing polynomial
has low degree.Comment: 22 page
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
Analytic combinatorics studies the asymptotic behaviour of sequences through
the analytic properties of their generating functions. This article provides
effective algorithms required for the study of analytic combinatorics in
several variables, together with their complexity analyses. Given a
multivariate rational function we show how to compute its smooth isolated
critical points, with respect to a polynomial map encoding asymptotic
behaviour, in complexity singly exponential in the degree of its denominator.
We introduce a numerical Kronecker representation for solutions of polynomial
systems with rational coefficients and show that it can be used to decide
several properties (0 coordinate, equal coordinates, sign conditions for real
solutions, and vanishing of a polynomial) in good bit complexity. Among the
critical points, those that are minimal---a property governed by inequalities
on the moduli of the coordinates---typically determine the dominant asymptotics
of the diagonal coefficient sequence. When the Taylor expansion at the origin
has all non-negative coefficients (known as the `combinatorial case') and under
regularity conditions, we utilize this Kronecker representation to determine
probabilistically the minimal critical points in complexity singly exponential
in the degree of the denominator, with good control over the exponent in the
bit complexity estimate. Generically in the combinatorial case, this allows one
to automatically and rigorously determine asymptotics for the diagonal
coefficient sequence. Examples obtained with a preliminary implementation show
the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201
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
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