On the classification of planar monomials over fields of square order (Q765804)

Let \(\mathbb F_q\) be a finite field of characteristic \(p\). A polynomial \(f\in \mathbb F_q[x]\) is called \textit{planar} if, for every non-zero \(a\in\mathbb F_q\), \(f(x+a)-f(x)\) is a permutation polynomial over \(\mathbb F_q\). The main conjecture (derived from a suggestion of \textit{P. Dembowski} and \textit{T. G. Ostrom} [Math. Z. 103, 239--258 (1968; Zbl 0163.42402)]) is that a planar polynomial must have the form \(\sum a_{ij}x^{p^i+p^j}\). The conjecture is known to hold if \(q=p\), for monomials if \(q=p^2\) and it fails for \(p=3\). Here the authors prove the conjecture for monomials when \(q=p^4\) and \(p\geq 5\). They also give a general result which should help settle the conjecture for monomials when \(q=p^{2^k}\). Specifically, let \(q=p^e\), \(p\) odd and \(e=2w\), \(w\geq 2\). If \(x^n\) is planar (\(n<q\)) then \(n\equiv 2p^j\pmod{p^w-1}\), where \(j<w\), and there are only three choices for the base \(p\) expansion of \(n\).