On the graph of a function in many variables over a finite field, Des

Abstract. Some improved bounds on the number of directions not determined by a point set in the affine space AG(k, q) are presented. More precisely, if there are more than pe(q−1) directions not determined by a set of qk−1 points S then every hyperplane meets S in 0 modulo pe+1 points. This bound is shown to be tight in the case pe = qs and when q = pes sets of qk−1 points that do not meet every hyperplane in 0 modulo pe+1 points and have a little less than pe(q − 1) non-determined directions are constructed. 1