Permutation Binomials over Finite Fields

Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for which the binomial $x^n(x^{(q-1)/r} + a)$ is a permutation polynomial in the cases $r = 2$ and $r = 3$.Comment: 13 page

Permutation Polynomials with Carlitz Rank 2

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. The Carlitz rank of a permutation polynomial is a important measure of complexity of the polynomial. In this paper we find the sharp lower bound for the weight of any permutation polynomial with Carlitz rank $2$, improving the bound found by G\'omez-P\'erez, Ostafe and Topuzo\u{g}lu in that case.Comment: 10 pages, comments are welcom