Counting arcs in projective planes via Glynn’s algorithm
| Autor: | Luke Peilen, Susie Kimport, Max Weinreich, Rachel Lawrence, Nathan Kaplan |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
51E20
51E15 51A35 Plane (geometry) 010102 general mathematics Order (ring theory) 0102 computer and information sciences Function (mathematics) 01 natural sciences Combinatorics Finite field 010201 computation theory & mathematics Line (geometry) FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) Geometry and Topology Projective plane 0101 mathematics Special case Algorithm Mathematics |
| Zdroj: | Journal of Geometry. 108:1013-1029 |
| ISSN: | 1420-8997 0047-2468 |
| DOI: | 10.1007/s00022-017-0391-1 |
| Popis: | An $n$-arc in a projective plane is a collection of $n$ distinct points in the plane, no three of which lie on a line. Formulas counting the number of $n$-arcs in any finite projective plane of order $q$ are known for $n \le 8$. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted $9$-arcs in the projective plane over a finite field of order $q$ and showed that this count is a quasipolynomial function of $q$. We present a formula for the number of $9$-arcs in any projective plane of order $q$, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as a special case. We obtain our formula from a new implementation of an algorithm due to Glynn; we give details of our implementation and discuss its consequences for larger arcs. 19 pages, to appear in Journal of Geometry |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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