Orchards in elliptic curves over finite fields |
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Institution: | 1. Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada;2. Mathematical and Physical Sciences division, School of Arts and Sciences, Ahmedabad University, Central Campus, Navrangpura, Ahmedabad 380009, India;1. Department of Mathematics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic;2. Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6;1. Department of Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic;2. Institut de Mathématiques de Toulon, campus La Garde, 83041 Toulon, France;1. CNRS, IMJ-PRG, Sorbonne Université, 4 Place Jussieu, F-75252 Paris Cedex 05, France;2. CNRS, IRMA, Université de Strasbourg, 7 rue René Descartes, F-67084 Strasbourg, France;3. Johann Radon Institute, RICAM, Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria |
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Abstract: | Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of 3-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of 3-rich lines agrees with the Green-Tao formula. |
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Keywords: | Point-line arrangements Orchard problem Elliptic curves Group law Application of finite fields |
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