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Semiovals contained in the union of three concurrent lines. (English) Zbl 1134.51007

A semioval in a projective plane \(\mathcal P\) of order \(q\) is a non-empty set \(S\) of points such that for every \(p\in S\) there exists a unique line \(t_{p}\), called the tangent to \(S\) at \(p\), with \(S \cap t_{p} = \{ p \}\). In 2004, G. Kiss and J. Ruff gave a complete classification of semiovals which are contained in the sides of a triangle [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 47, 97–105 (2004; Zbl 1088.51002)]. In the present note the authors provide an algebraic description of a semioval and give a sharp upper bound on their size. The main part of the paper introduces strong semiovals and contains a classification of strong semiovals in PG\((2,p)\) and PG\((2,p^2)\) for any odd prime \(p\).

MSC:

51E21 Blocking sets, ovals, \(k\)-arcs

Citations:

Zbl 1088.51002
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References:

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