An algorithm for counting arcs in higher-dimensional projective space

2021/08/02 by Kelly Isham, Isham, Kelly · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2108.01024

openalex publication_date 2021/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An n arc in (k-1)-dimensional projective space is a set of n points so that no k lie on a hyperplane. In 1988, Glynn gave a formula to count n-arcs in the projective plane in terms of simpler combinatorial objects called superfigurations. Several authors have used this formula to count n-arcs in the projective plane for n ≤ 10. In this paper, we determine a formula to count n-arcs in projective 3-space. We then use this formula to give exact expressions for the number of n-arcs in ℙ³(\mathbbF_q) for n ≤ 7, which are polynomial in q for n ≤ 6 and quasipolynomial in q for n=7. Lastly, we generalize to higher-dimensional projective space.

Cited by

• Configurations of 10 points and their incidence varieties

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