  Motivated by many recent constructions of permutation polynomials over
$\mathbb{F}_{q^2}$, we study permutation polynomials over $\mathbb{F}_{q^3}$ in
terms of their coefficients. Based on the multivariate method and resultant
elimination, we construct several new classes of sparse permutation polynomials
over $\mathbb{F}_{q^3}$, $q=p^{k}$, $p\geq3$. Some of them are complete
mappings.
