Planar Polynomials arising from Linearized polynomials

Abstract: In this paper we construct planar polynomials of the type $f_{A,B}(x)=x(x^{q^2}+Ax^{q}+Bx)\in \mathbb{F}{q^3}[x]$, 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.