Group realizations of configurations
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Date
2017
Authors
Lanyon, Jeffrey
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Abstract
One of the basic questions in this topic is about the "existence" and the related "realization" of abstract configurations. Motivated by the fact that a non-singular cubic curve in the projective plane over a field admits a geometrically defined group law "⊕" such that three points {P, Q, R} are collinear if and only if the sum P⊕Q⊕R = 0 under the group law, we define the concept of group realization of a given (nk) configuration. Group realizations are in turn used to construct geometric realizations (of (n3)'s and (n4)'s) in the real or the complex projective plane. Using a variety of techniques from algebra and number theory like the resultant of polynomials, Hensel's Lemma on lifting primitive roots to prime powers, companion matrices, and Bunyakovsky's conjecture we prove the existence of infinitely many realizable configurations, including:
1. Realizations of cyclic (n3) configurations using the group structure on cubic curves
over the real or the complex plane;
2. Realizations of cyclic (n4) configurations using the group structure on non-circular
ellipses where now blocks of 4 points are circles over the real plane.
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Keywords
mathematics, geometry, mathematical configurations, group structure