Counting arcs in projective planes via Glynn’s algorithm

被引：3

作者：

Kaplan N. ^{[1
]}

Kimport S. ^{[2
]}

Lawrence R. ^{[3
]}

Peilen L. ^{[3
]}

Weinreich M. ^{[3
]}

机构：

[1] Department of Mathematics, University of California, Irvine, 92697, CA

[2] Department of Mathematics, Stanford University, Stanford, 94305, CA

[3] Department of Mathematics, Yale University, New Haven, 06511, CT

来源：

Journal of Geometry | 2017年 / 108卷 / 3期

基金：

美国国家科学基金会;

关键词：

Arcs; configurations of points and lines; finite projective planes; incidence structures; linear spaces; non-Desarguesian projective planes;

D O I：

10.1007/s00022-017-0391-1

中图分类号：

学科分类号：

摘要：

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≤ 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. © 2017, Springer International Publishing.

引用

页码：1013 / 1029

页数：16

共 17 条

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[10] Kaplan N., Kimport S., Lawrence R., Peilen L., Weinreich, M.: The number of 10 -arcs in the projective plane is not quasipolynomial (in preparation), (2017)

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