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.

翻译：平面函数由Dembowski和Ostrom提出，是从有限域到自身的函数，能够构造有限射影平面。然而，此类函数仅存在于奇特征有限域中。近十年来，因其在理论上的重要性以及在各领域中的深刻多样应用，平面函数备受关注。本文聚焦于三次扩域$\mathbb{F}_{q^3}/\mathbb{F}_q$上的平面函数，具体研究了$\mathbb{F}_{q^3}$上形如$\sum_{0\le i\le j<3}a_{ij}x^{q^i+q^j}$的二项式和三项式平面多项式，完整刻画了这些多项式，并确定了其等价类以实现分类。我们的成果基于代数射影曲线与有限域上经典代数工具的联系。