On a generalization of planar functions

Planar functions have deep applications in different areas of mathematics. We give a generalization of planar functions, and we obtain construction and classification results concerning these new objects. More in detail, for a prime power \(q=p^r\) and \(\beta \in {\mathbb {F}}_q{\setminus }\{0,1\}\), we call a function \(f : {\mathbb {F}}_q \rightarrow {\mathbb {F}}_q\)\(\beta \)-planar function in \({\mathbb {F}}_{q}\) if for each \(\gamma \in {\mathbb {F}}_{q}\) we have that \(f(x+\gamma )-\beta f(x)\) permutes \({\mathbb {F}}_{q}\). In this work, we study \(\beta \)-planar monomials: We provide some necessary conditions for general r, whereas for \(r\le 3\) we construct examples of \(\beta \)-planar monomials. Connections with algebraic curves are used to prove nonexistence results.