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On sets defining few ordinary hyperplanes

Lin, Aaron and Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2020) On sets defining few ordinary hyperplanes. Discrete Analysis. ISSN 2397-3129

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Identification Number: 10.19086/da.11949

Abstract

Let P be a set of n points in real projective d-space, not all contained in a hyperplane, such that any d points span a hyperplane. An ordinary hyperplane of Pis a hyperplane containing exactly d points of P. We show that if d > 4, the numberof ordinary hyperplanes of P is at least􀀀n􀀀1d􀀀1􀀀 Od(nb(d􀀀1)=2c) if n is sucientlylarge depending on d. This bound is tight, and given d, we can calculate the exactminimum number for suciently large n. This is a consequence of a structuretheorem for sets with few ordinary hyperplanes: For any d > 4 and K > 0, ifn > CdK8 for some constant Cd > 0 depending on d, and P spans at most K􀀀n􀀀1d􀀀1ordinary hyperplanes, then all but at most Od(K) points of P lie on a hyperplane,an elliptic normal curve, or a rational acnodal curve. We also nd the maximumnumber of (d+1)-point hyperplanes, solving a d-dimensional analogue of the orchardproblem. Our proofs rely on Green and Tao's results on ordinary lines, our earlierwork on the 3-dimensional case, as well as results from classical algebraic geometry.

Item Type: Article
Official URL: https://discreteanalysisjournal.com/about
Additional Information: © 2020 The Authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 10 Feb 2020 10:18
Last Modified: 01 Aug 2020 23:39
URI: http://eprints.lse.ac.uk/id/eprint/103320

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