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Planar functions and planes of Lenz-Barlotti class II. (English) Zbl 0872.51007
Planar functions were introduced by Dembowski and the reviewer [P. Dembowski and T. G. Ostrom, Math. Z. 103, 239-258 (1968; Zbl 0163.42402)]. To take the simplest case, let \(F= {\mathcal G} F(p^r)\). A function from \(F\) onto \(F\) is planar if the mapping \(x\to f(x+ m)- f(x)\) is \(1-1\) onto for each \(m\neq 0\).
If \(f\) is such a planar function there exists an affine plane whose points are ordered pairs \((x, y)\) of elements of \(F\) and whose lines are sets of points \((x,y)\) such that \(y= f(x+m) +b\) one line for each \(m,b\). For each \(c\in F\) there is also a line consisting of all points such that \(x=c\). (A nicer representation is to take \(y= f(x+m)- f(x)- f(m) +b)\). If \(f(X)= \sum a_{ij} X^{p^i+ p^j}\), the plane turns out to be a semi-field plane. The authors show that if \(p=3\), that \(x^k\) is planar function if \(k\) has the form \({1\over 2} (3^\alpha+2)\).
Furthermore, the projective version of the plane is \((p,L)\) transitive for just one choice of \(p\) and \(L\). (In the natural coordinatization, \(p= (\infty)\) and \(L= L_\infty)\).

51E15 Finite affine and projective planes (geometric aspects)
51A35 Non-Desarguesian affine and projective planes
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