Let φ(x) be an Eisenstein polynomial of degree n over a local field and α be a root of φ(x). Our main tool is the ramification polygon of φ(x), that is the Newton polygon of ρ(x) = φ(αx+α)/(αx). We present a method for determining the Galois group of φ(x) in the case where the ramification polygon consists of one segment.

Given a simple n-vertex polygon, the triangulation problem is to partition the interior of the polygon into n-2 triangles by adding n-3 nonintersecting diagonals. We propose an O(n log logn)-time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that triangu-lation is not as hard as sorting. Improved algorithms for several other computational geometry...

We present a diagram that captures containment information for scalable rotated and translated versions of a convex polygon. For a given polygon P and a contact point q in a point set S, the diagram parameterizes possible translations, rotations, and scales of the polygon in order to represent containment regions for each additional point v in S. We present geometric and combinatorial propertie...

The geodesic k-center problem in a simple polygon with n vertices consists in the following. Find a set S of k points in the polygon that minimizes the maximum geodesic distance from any point of the polygon to its closest point in S. In this paper, we focus on the case where k = 2 and present an exact algorithm that returns a geodesic 2-center in O(n log n) time.

We provide the rst tight bound for covering a polygon with n vertices and h holes with vertex guards. In particular, we provide tight bounds for the number of oodlights, placed at vertices or on the boundary , suucient to illuminate the interior or the exterior of an orthogonal polygon with holes. Our results lead directly to simple linear, and thus optimal, algorithms for computing a covering ...

Instead of using the polygon defined by adjacent vertices to a vertex (called the ball) or its kernel [1], we propose a modified polygon that is easy to compute, convex and an approximation of the kernel. We call this polygon the “quick kernel ball region.” This novel algorithm is presented in details. It is easy to implement and effective in constraining a vertex to remain within its feasible ...

We consider a model consisting of a self-avoiding polygon occupying a variable density of the sites of a square lattice. A fixed energy is associated with each 90-bend of the polygon. We use a grand canonical ensemble, introducing parameters μ and β to control average density and average (total) energy of the polygon, and show by Monte Carlo simulation that the model has a first order, nematic ...

Given a simple polygon P , we consider the problem of finding a convex polygon Q contained in P that minimizes H(P, Q), where H denotes the Hausdorff distance. We call such a polygon Q a Hausdorff core of P . We describe polynomial-time approximations for both the minimization and decision versions of the Hausdorff core problem, and we provide an argument supporting the hardness of the problem.

We introduce the notion of free polygons as combinatorial building blocks for convex integral polygons; that is, polygons with vertices having integer coordinates. In this context, an Euler-type formula is derived for the number of integer points in the interior of an integral polygon. This leads in turn to a formula for the area of an integral polygon P via the enumeration of free integral tri...