On the number of slopes of the graph of a function defined on a finite field
Let \(U\) be a set of \(q\) points in the affine plane \(AG(2,q)\), \(q=p^n\), \(p\) any prime. In recent years several authors have studied the following problem: how many directions are determined by the set \(U\)? The answer to this question is connected to blocking sets theory [e.g. \textit{A. Blokhuis}, Bolyai Soc. Math. Stud. 2, 133-155 (1996; Zbl 0849.51005)] and to permutation polynomials theory [e.g. \textit{R. J. Evans}, \textit{J. Green}, and \textit{H. Niederreiter}, Mich. Math. J. 39, No. 3, 405-413 (1992; Zbl 0777.11052)]. In this paper the authors determine the possibilities for the number \(N\) of directions of secants of \(U\) and, if \(N<{q+3\over 2}\), they characterize the set \(U\). Precisely, they prove that, if \(e\) (with \(0\leq e\leq n)\) is the largest integer such that each secant of \(U\) meets it in a multiple of \(p^e\) points, then one of the following cases occurs: (i) \(e=0\) and \({q+3 \over 2}\leq N\leq q+1\); (ii) \(e=1\), \(p=2\), and \({q+5\over 3}\leq N\leq q-1\); (iii) \(p^e>2\), \(e\mid n\), and \({q\over p^e} +1\leq N\leq {q-1 \over p^e-1}\); (iv) \(e=n\) and \(N=1\). Moreover, if \(p^e>3\) or \((p^e=3\) and \(N=q/3+1)\), and \(U\) contains the origin, then \(U\) is a \(GF(p^e)\)-linear subspace of \(AG (2,q)\). Except for the lower bound in case (ii), all bounds given for \(N\) are sharp.
scientific article; zbMATH DE number 1278835