Tópicos recentes envolvendo polinômios de permutação sobre corpos finitos

Abstract

In this work, we will present some current topics involving permutation polynomials over finite fields. In particular, we will show the necessary theory to count the number of permutation binomials of the forms $x^n(x^{\frac{q-1}{2}}+a)$ and $x^n(x^{\frac{q-1}{3}}+a)$. We will also present the Carlitz rank concept for permutation polynomials and a new lower bound for the number of nonzero coefficients of permutation polynomials with rank $2$.