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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 $$L= L_\infty)$$.

##### MSC:
 51E15 Finite affine and projective planes (geometric aspects) 51A35 Non-Desarguesian affine and projective planes
##### Keywords:
permutation polynomials; planar function; affine plane
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