On likeable translation planes of even order
A function f: GF(q)\(\to GF(q)\), where q is not a power of 3, is called ''likeable'' if (i) f is additive and (ii) the equation \(x^ 2=xy^ 2- \frac{1}{3}y^ 4+yf(y)\) has only \((x,y)=(0,0)\) as a solution. Using such a function one may construct a translation plane, which is also called likeable. The concept of a likeable function was introduced by W. M. Kantor [Isr. J. Math. 42, 227-234 (1982; Zbl 0511.51011)]. He observed that when \(q=2^ r\) (r odd, \(r\geq 3)\) the function \(f(y)=c^ 2y+cy^ 2 (c\in GF(2^ r))\) is likeable and that the planes obtained from different values of c are all isomorphic. The plane obtained is the so- called Betten-Walker plane of order \(2^{2r}\) [the reviewer, Geom. Dedicata 2, 327-339 (1973; Zbl 0272.50028), and \textit{M. Walker}, Geom. Dedicata 5, 135-146 (1976, Zbl 0356.50022)]. The author proves this Theorem: The only likeable translation plane of even order \(2^{2r}\) is the Betten-Walker plane, which occurs if and only if r is odd and \(r\geq 3\).
scientific article; zbMATH DE number 3848117