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

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 p ^e (q − 1) directions not determined by a set of q ^k-1 points then every hyperplane meets in 0 modulo p ^e+1 points. This bound is shown to be tight in the case p ^e = q ^s and when q = p ^es sets of q ^k-1 points that do not meet every hyperplane in 0 modulo p ^e+1 points and have a little less than p ^e (q − 1) non-determined directions are constructed. The author acknowledges the support of the Ramon y Cajal programme and the project MTM2005-08990-C02-01 of the Spanish Ministry of Science and Education and the project 2005SGR00256 of the Catalan Research Council.