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.