On the graph of a function in two variables over a finite field (Q2494275)

Let \(AG(3,q)\) and \(PG(3,q)\) denote respectively the affine and the projective 3-dimensional space over the Galois field \(GF(q)\). Any function \(f\) from \(GF(q)^2\) to \(GF(q)\) determines a set \[ {\mathcal W}=\{ (a,b,f(a,b)): a,b \in GF(q) \}. \] The set of directions determined by \(f\) (or by \({\mathcal W}\)) is the set of points obtained as the intersection of the plane \(x_3=0\) with any line joining two distinct points of \({\mathcal W}\). The main result of the paper under review is the following theorem. Theorem 2.2. Let \({\mathcal W} \subset AG(3,q) \subset PG(3,q) , q=p^h\), \(p\) prime, \(| {\mathcal W} | =q^2\). If the number of directions not determined by \({\mathcal W}\) is at least \(q\), then every plane of \(PG(3,q)\) meets \({\mathcal W}\) in 0 (mod \(p\)) points. More specific results are proven under supplementary hypotheses. The authors apply these results to the ovoids of generalized quadrangles.