157,562 research outputs found

    Maximal integral point sets over Z^2

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    Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P = {p1,..., pn} ⊂ Z² a maximal integral point set over Z 2 if all pairwise distances are integral and every additional point pn+1 destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle

    Maximal integral point sets in affine planes over finite fields

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    Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane Fq2\mathbb{F}_q^2 over a finite field Fq\mathbb{F}_q, where the formally defined squared Euclidean distance of every pair of points is a square in Fq\mathbb{F}_q. It turns out that integral point sets over Fq\mathbb{F}_q can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over Fq\mathbb{F}_q for q≤47q\le 47. Furthermore, we give two series of maximal integral point sets and prove their maximality.Comment: 18 pages, 3 figures, 2 table

    Integral point sets over finite fields

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    We consider point sets in the affine plane Fq2\mathbb{F}_q^2 where each Euclidean distance of two points is an element of Fq\mathbb{F}_q. These sets are called integral point sets and were originally defined in mm-dimensional Euclidean spaces Em\mathbb{E}^m. We determine their maximal cardinality I(Fq,2)\mathcal{I}(\mathbb{F}_q,2). For arbitrary commutative rings R\mathcal{R} instead of Fq\mathbb{F}_q or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.Comment: 22 pages, 4 figure

    Volume computation for polytopes and partition functions for classical root systems

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    This paper presents an algorithm to compute the value of the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes. As an application, we present an efficient computation of the partition function for classical root systems.Comment: 55 pages, 14 figures. Maple programs available at http://www.math.polytechnique.fr/~vergne/work/IntegralPoints.htm
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