On Schoenberg's Rational Polygon Problem

On the Erdos-Szekeres convex polygon problem

Let ES(n) be the smallest integer such that any set of ES(n) points in the plane in general position contains n points in convex position. In their seminal 1935 paper, Erdős and Szekeres showed that ES(n) ≤ ( 2n−4 n−2 ) + 1 = 4. In 1960, they showed that ES(n) ≥ 2 + 1 and conjectured this to be optimal. In this paper, we nearly settle the Erdős-Szekeres conjecture by showing that ES(n) = 2.

The Polygon Exploration Problem

We present an on-line strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed off-line. Our analysis is doubly founded on a novel geometric structure called the angle hull. Let D be a connected region inside a simple polygon, P . We define the angl...

Points at Rational Distance from the Vertices of a Unit Polygon

On the Canonical k-vertex Polygon Spatial Retrieval Problem

The polygon retrieval problem on points is the problem of preprocessing a set of n points on the plane, so that given a polygon query, the subset of points lying inside it can be reported efficiently. It is of great interest in areas such as Computer Graphics, CAD applications, Spatial Databases and GIS developing tasks. In this paper we study the problem of canonical k-vertex polygon queries o...

The Newton Polygon of Plane Curves with Many Rational Points

This curve has the points (1 : 0 : 0) and (0 : 1 : 0) at infinity over any field. The affine equation is XY +Y +X = 0. The origin is a point of this curve. If (x, y) ∈ F8 is a point of this curve with nonzero coordinates, then x = 1. So 0 = xy + y + x = xy + xy + x = x[(xy) + (xy) + 1]. Let t = xy. Then t + t+ 1 = 0. So the Klein quartic has 3.7 = 21 rational points over F8 with nonzero coordin...