On the graph of a function in many variables over a finite field
The paper contains some improved bounds on the number of directions not determined by a point set in the affine space \(AG(k, q)\). Let \(x\) and \(y\) be two distinct points of \(AG(k, q)\), the \textit{direction} determined by \(x\) and \(y\) is the point \(d= \langle x - y \rangle = \{\lambda(x-y) \mid \lambda \in GF(q)\}\) of \(PG(k-1, q)\). The following result, which improves a previous result on the same topic, is proved. Theorem. Let \(q = p^h\) and \(1\leq p^e\leq q^{k-2}\), where \(e\) is a non-negative integer. If there are more than \(p^e(q-1)\) directions not determined by a set \(S\) of \(q^{k-1}\) points in \(AG(k, q)\) then every hyperplane meets \(S\) in \(0\) modulo \(p^{e+1}\) points. Constructions of point-sets which meet the bound in the case \(p^e = q^s\) and almost reach the bound when \(q= p^e\) are given in Section \(3\), and the paper ends with a section in which three conjectures arising from the above theorem (\textit{cylinder conjectures}) are stated.
scientific article; zbMATH DE number 5536726
scientific article