On the number of directions determined by a point set in AG(2,p) Original Research Article

It has been known for a long time that a p-element point set in AG(2,p), which is not a line, determines at least (p+3)/2 directions (Rédei, Lückenhafte Polynome über endlichen Körpern, Birkhauser Verlag, Basel, 1970 (English translation: Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973)). In this paper we look for sets determining more than (p+3)/2 directions. We prove that besides two examples no set determines (p+5)/2 directions, give an infinite series of examples determining 7p/9 directions approximately and prove results about the graph of monomials. These results suggest a conjecture, namely that no point set can determine N directions with (p+3)/2