Characterization of complete exterior sets of conics
Let \(q\) be an odd prime power, and let \(\pi=PG(2,q)\) denote the desarguesian projective plane of order \(q\). A complete exterior set is defined to be a set of \((q+1)/2\) exterior points with respect to a fixed nondegenerate conic of \(\pi\) such that any two of these points determine a passant line. (Of course, the maximum size of any set of such points is \((q+1)/2.)\) Clearly, the \((q+1)/2\) exterior points on any passant line form a complete exterior set. In the paper under review the authors show that these are the only possible examples when \(q\equiv 1\pmod 4\). For \(q=7\), 11, 19, 23, 27 and 31 nonlinear examples of complete exterior sets are found (exhaustively by computer). However, it is conjectured that for \(q>31\) the only examples of complete exterior sets are the linear ones. The main idea behind the proof in the \(q\equiv 1\pmod 4\) case is to carefully coordinatize so that the problem may be reformulated as follows: characterize all bijections \(\Phi:GF(q)^*\to GF(q)^*\) of order 2, sending squares to nonsquares, such that for any two squares \(s\) and \(t\) we have \((s-t)(s-\Phi(t))(\Phi(s)-t)(\Phi(s)-\Phi(t))\) is a nonsquare. The bijections of the form \(\Phi(x)=n/x\) for some fixed nonsquare \(n\) in \(GF(q)\) correspond to the linear complete exterior sets. To show that these are the only possible bijections when \(q\equiv 1\pmod 4\) the authors cleverly use another reformulation of the problem in terms of the Paley graph as well as the following consequence of a theorem of \textit{L. Carlitz} and \textit{R. McConnel} ['A theorem on permutations in a finite field', Proc. Am.Math. Soc. 11, 456-459 (1960)]: Let \(f:GF(q)\to GF(q)\) be a function with the property that for all \(x,y\in GF(q)\) with \(x\neq y\), \((f(x)-f(y))/(x-y)\) is a nonsquare. Then \(f\) is of the form \(f(x)=a+bx^{p^ j}\), where \(a,b\in GF(q)\) with \(b\) a nonsquare and \(p\) is the characteristic of the field.
scientific article; zbMATH DE number 89339