On small complete arcs in a finite plane
type: Item
instance of: scholarly article
title: On small complete arcs in a finite plane
zbMATH Open document ID: 0891.51006
DOI: 10.1016/S0012-365X(96)00315-9
published in: Discrete Mathematics
publication date: 1998-06-12T00:00:00Z
review text: A set \(K\) of points in the finite projective plane \(\text{ PG}(2,q)\) is called an arc if every line intersects \(K\) in at most two points. An arc \(K\) is called complete if \(K\) cannot be extended to a larger arc. \textit{A. Blokhuis} [Bolyai Soc. Math. Stud. 2, 133-155 (1996; Zbl 0849.51005)] mentions that any complete arc in \(\text{ PG}(2,p)\), \(p\) prime, has at least \(\sqrt {3p}\) points. In the paper under review the author both enhances and generalizes this result by showing that any complete arc in \(\text{ PG}(2,q)\), \(q=p\) or \(q=p^2\), has more than \(\sqrt {3q} + {1\over2}\) points. The proof uses the theory of lacunary polynomials. A polynomial is called lacunary if it has a long run of zeros in its sequence of coefficients.
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 1110071
zbMATH Keywords: small complete arc
zbMATH Keywords: blocking set
author: Simeon Ball
MaRDI profile type: Publication
cites work: Q4878658
cites work: On the size of a blocking set in \(\text{PG}(2,p)\)
cites work: Q3851217
cites work: New examples of complete k-arcs in PG(2,q)
cites work: Q5669405
description: scientific article; zbMATH DE number 1110071