Characterizations of analytic planar functions
Let \(G\) and \(H\) be abelian groups. A function \(f: G\to H\) is called planar if for all \(d\in G\setminus \{0\}\) the function \(f_d: G\to H: x\mapsto f(x+ d)- f(x)\) is bijective. With every planar function one can associate an affine plane by taking as points the elements of \(G\times H\) and as lines all translates of the graph of \(f\) and all sets \(\{c\}\times H\) with \(c\in G\). Assume now that both \(G\) and \(H\) are equal to the additive group of the complex number field and that \(f\) is complex analytic. The author proves that then \(f\) is necessarily a quadratic polynomial and the resulting plane is desarguesian. An analogous result for the case that \(G\) and \(H\) are isomorphic to the additive group of a finite prime field has been proved independently by Gluck, Hiramine, and Rónyai-Szönyi.
scientific article; zbMATH DE number 802363
scientific article