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Necessary and sufficient conditions on $A,B\in \mathbb{F}_{q^3}^*$ for$f(X)=X^{q^2-q+1}+AX^{q^2}+BX$ being a permutation polynomial of$\mathbb{F}_{q^3}$ are investigated via a connection with algebraic varietiesover finite fields.

On Permutation Trinomials of the type $X^{q^2-q+1}+AX^{q^2}+BX$ over $\mathbb{F}_{q^3}$