Permutation polynomials and complete permutation polynomials over $\mathbb{F}_{q^{3}}$

Abstract: 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.