Abstract. In the affine plane $AG(2,q)$ over the Galois field $GF(q)$, $q=p^h$ and $p$ an odd prime, let the parabola ${\cal P}$ be given in its canonical form $y=x^2$. For a positive integer $v< h$, put $d=p^v$ and denote by $K_d$ the set of size $q/d$ consisting of all points $(x^d-x,(x^d-x)^2)$ with $x$ ranging over $GF(q)$. The set $K_d$ lies on ${\cal P}$, and hence $K_d$ may be viewed as a $q/d$-gon in ${\cal A}$, because no three points of $K_d$ are collinear. Szőnyi [7] proved that if $q/d$ is sufficiently large, then the chords of $K_d$ cover every affin point apart from those in ${\cal P} \setminus K_d$. In this paper we give an upper bound for the number $n_P(d)$ of chords of $K_d$ through a given point $P$ outside of ${\cal P}$.

AMS Subject Classification (1991): 51E20, 51E21

Received October 24, 1995 and in revised form March 5, 1996. (Registered under 5727/2009.)