Related papers: The W. Thurston Algorithm Applied to Real Polynomi…

The dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal…

The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric…

We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of…

We determine a strong form of the decomposition theorem for proper toric maps over finite fields.

The aim of this paper is to show that there exists a deterministic algorithm that can be applied to compute the factors of a polynomial of degree 2, defined over a finite field, given certain conditions.

We demonstrate that the question whether or not a given postcritically finite topological ramified covering map of the 2-sphere is Thurston equivalent to a rational map is algorithmically decidable.

We extend Thurston's combinatorial criterion for postcritically finite rational maps to a class of rational maps with bounded type Siegel disks. The combinatorial characterization of this class of Siegel rational maps plays a special role…

We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e. with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real…

In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…

We describe the topology of a general polynomial mapping $f:\Bbb C^2\to\Bbb C^2.$

We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…

The paper studies constructions of irreducible polynomials over finite fields using polynomial composition method.

We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and…

This note will study complex polynomial maps of degree $n\ge 2$ with only one critical point.

We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies…

This paper describes a method for computing all F-pure ideals for a given Cartier map of a polynomial ring over a finite field.

Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as…

In order to perform numerical studies of long-term stability in nonlinear Hamiltonian systems, one needs a numerical integration algorithm which is symplectic. Further, this algorithm should be fast and accurate. In this paper, we propose…

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that…

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role…