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

Mar 2023

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

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