26 research outputs found
Irredundant Triangular Decomposition
Triangular decomposition is a classic, widely used and well-developed way to
represent algebraic varieties with many applications. In particular, there
exist sharp degree bounds for a single triangular set in terms of intrinsic
data of the variety it represents, and powerful randomized algorithms for
computing triangular decompositions using Hensel lifting in the
zero-dimensional case and for irreducible varieties. However, in the general
case, most of the algorithms computing triangular decompositions produce
embedded components, which makes it impossible to directly apply the intrinsic
degree bounds. This, in turn, is an obstacle for efficiently applying Hensel
lifting due to the higher degrees of the output polynomials and the lower
probability of success. In this paper, we give an algorithm to compute an
irredundant triangular decomposition of an arbitrary algebraic set defined
by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant
triangular decomposition, we were able to give intrinsic degree bounds for the
polynomials appearing in the triangular sets and apply Hensel lifting
techniques. Our decomposition algorithm is randomized, and we analyze the
probability of success
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
Real Root Isolation of Regular Chains
We present an algorithm RealRootIsolate for isolating the real roots of a system of multivariate polynomials given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation, combining a symbolic object, namely a regular chain, and a numerical approximation given by intervals. Our isolation algorithm is a generalization, for regular chains, of the algorithm proposed by Collins and Akritas. We have implemented RealRootIsolate as a command of the module SemiAlgebraicSetTools of the RegularChains library in Maple. Benchmarks are reported.
Global Identifiability of Differential Models
Many real-world processes and phenomena are modeled using systems of ordinary
differential equations with parameters. Given such a system, we say that a
parameter is globally identifiable if it can be uniquely recovered from input
and output data. The main contribution of this paper is to provide theory, an
algorithm, and software for deciding global identifiability. First, we
rigorously derive an algebraic criterion for global identifiability (this is an
analytic property), which yields a deterministic algorithm. Second, we improve
the efficiency by randomizing the algorithm while guaranteeing the probability
of correctness. With our new algorithm, we can tackle problems that could not
be tackled before. A software based on the algorithm (called SIAN) is available
at https://github.com/pogudingleb/SIAN