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Tue, 15 Apr 2008 11:07:00 +0200Tue, 15 Apr 2008 11:07:00 +0200Integral point sets over Z_n^m
http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/377
There are many papers studying properties of point sets in the Euclidean space or on integer grids, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of Z_n, and study the properties of the resulting combinatorial structures.Axel Kohnert; Sascha Kurzpreprinthttp://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/377Tue, 15 Apr 2008 11:07:00 +0200On the minimum diameter of plane integral point sets
http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/375
Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2,n) of a plane integral point set consisting of n points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2,n) achieving the known upper bound n^{c_2loglog n} up to a constant in the exponent.Sascha Kurz; Alfred Wassermannpreprinthttp://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/375Tue, 15 Apr 2008 11:01:24 +0200Inclusion-maximal integral point sets over finite fields
http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/372
We consider integral point sets in affine planes over finite fields. Here an integral point set is a set of points in $GF(q)^2$ where the formally defined Euclidean distance of every pair of points is an element of $GF(q)$. From another point of view we consider point sets over $GF(q)^2$ with few and prescribed directions. So this is related to Redeis work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces. In this article we study the spectrum of integral point sets over $GF(q)^2$ which are maximal with respect to inclusion. We give some theoretical results, constructions, conjectures, and some numerical data.Michael Kiermaier; Sascha Kurzpreprinthttp://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/372Tue, 15 Apr 2008 10:52:18 +0200Maximal integral point sets over Z^2
http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/369
Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P={p(1),...,p(n)} in Z^2 a maximal integral point set over Z^2 if all pairwise distances are integral and every additional point p(n+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.Sascha Kurz; Andrey Radoslavov Antonovpreprinthttp://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/369Tue, 15 Apr 2008 10:45:28 +0200On the characteristic of integral point sets in $\mathbb{E}^m$
http://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/175
We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to provide new exact values for the minimum diameter of integral point sets.Sascha Kurzpreprinthttp://opus4.kobv.de/opus4-ubbayreuth/frontdoor/index/index/docId/175Mon, 05 Dec 2005 10:33:52 +0100