Planar Polynomials arising from Linearized polynomials

Abstract

In this paper we construct planar polynomials of the type $f_{A,B}(x)=x(x^{q^2}+A{x}^q+Bx)\in {\mathbb{F}}_{q^3}[x]$, with $A,B\in {\mathbb{F}}_q$. In particular we completely classify the pairs $(A,B)\in {\mathbb{F}}_q^2$ such that $f_{A,B}(x)$ is planar using connections with algebraic curves over finite fields.