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 characteristics. 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 trinomials over cubic and quartic extensions of finite fields. Our achievements are obtained using connections with quadratic forms and classical algebraic tools over finite fields. Furthermore, given the generality of our approach, the methodology presented could be employed to drive more planar functions on some finite extension fields.

翻译：平面函数由Dembowski和Ostrom引入，是从有限域到其自身的函数，能构造有限射影平面。然而，这类函数仅存在于奇特征有限域中。近十年来，因其理论价值及在多个领域的深远应用，平面函数引起了广泛关注。本文聚焦于有限域三次和四次扩域上的三进制平面三项式。我们通过二次型与经典有限域代数工具的关联取得研究成果。此外，基于方法的普适性，本文提出的方法论可用于推导更多有限扩域上的平面函数。