On finite planar functions, their proper shift planes and their derived translation planes (Q1411426)

Let \(F\) be a field. A function \(P : F^ k \rightarrow F^ k\) is called planar if \(x\mapsto P(x+a)-P(x)\) is bijective for every \(a \in F^ k \setminus \{0\}\). To every planar function one can assign an affine plane, the so--called shift plane corresponding to \(P\), whose point space is \(F^ n \times F^ n\) and whose lines are described by the equations \(y = P(x+a)+b\) and \(x=c\), respectively. Examples of planar functions are the maps \(P : \text{GF}(3^ e) \rightarrow \text{GF}(3^ e);\; x\mapsto x^n\), where \(n = (3^ \alpha+1)/2\) with \((\alpha,e)=1\), \(\alpha\) odd and \(\alpha\not\equiv\pm 1 \, (\text{mod}\, 2e)\). The corresponding shear planes are not desarguesian [\textit{R. S. Coulter} and \textit{R. W. Mathews}, Des. Codes Cryptography 10, 167--184 (1997; Zbl 0872.51007)]. If, in addition, \(e\) is even, then \(P\) can be considered as a planar function of \(F^ 2\), where \(F= \text{GF}(3^ {e/2})\). The authors prove that the formal Jacobian matrices of \(P : F^ 2 \rightarrow F^ 2\) yield a spread on \(F^ 4\) whose corresponding translation plane is desarguesian. The interesting fact is that the generating line \(y = P(x)\) of the shift plane is not an oval in this derived translation plane.