Classifying planar monomials over fields of order a prime cubed (Q2066387)

Consider \(\mathbb{F}_q\), a finite field with \(q\) elements. A polynomial \(f \in \mathbb{F}_q[x]\) is said to be a \textit{permutation polynomial} over \(\mathbb{F}_q\) if \(f\) induces a permutation over \(\mathbb{F}_q\). Moreover, we say that a polynomial \(f \in \mathbb{F}_q[x]\) is \textit{planar} if for every \(a \in \mathbb{F}_q^*\) the polynomial \(f(x+a)-f(x)\) is a permutation polynomial over \(\mathbb{F}_q\). In Theorem 1, the authors are able to prove that when \(q=p^3\) with \(p\) an odd prime, the monomial \(x^n\) is planar over \(\mathbb{F}_q\) if and only if \(n=p^i+p^j \pmod{q-1}\), with \(0\leq i,j<3\). In this way, they establish the Dembowski-Ostrom conjecture in this case. The proof uses Hermite's criteria and non-trivial manipulations.