On small complete arcs in a finite plane
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.
scientific article; zbMATH DE number 1110071