Binomials and Trinomials as Planar Functions on Cubic Extensions of Finite Fields

Planar functions, introduced by Dembowski and Ostrom, are functions from a finite field to itself that give rise to finite projective planes. They exist, however, only for finite fields of odd characteristic. They have attracted much attention in the last decade thanks to their interest in theory and those deep and various applications in many fields. This paper focuses on planar functions on a cubic extension $\mathbb F_{q^3}/\mathbb F_q$. Specifically, we investigate planar binomials and trinomials polynomials of the form $\sum_{0\le i\le j<3}a_{ij}x^{q^i+q^j}$ on $\mathbb F_{q^3}$, completely characterizing them and determine the equivalence class of those planar polynomials toward their classification. Our achievements are obtained using connections with algebraic projective curves and classical algebraic tools over finite fields.