CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

Let $p$ be a prime and $q$ a power of $p$. Let ${\mathbb{𝔽}}_q$ be the finite field with $q$ elements. A polynomial $f\in {\mathbb{𝔽}}_q\left[x\right]$ is called a permutation polynomial of ${\mathbb{𝔽}}_q$ if the associated mapping $x\mapsto f\left(x\right)$ from ${\mathbb{𝔽}}_q$ to ${\mathbb{𝔽}}_q$ is a permutation of ${\mathbb{𝔽}}_q$. Permutation polynomials over finite fields have important applications in coding theory, cryptography, finite geometry, combinatorics and computer science, among other fields. Recently, reversed Dickson polynomials over finite fields have been studied extensively by many for their general properties and permutation behaviour.

For $a\in {\mathbb{𝔽}}_q$, the $n$-th reversed Dickson polynomial of the $\left(k+1\right)$-th kind $D_{n,k}\left(a,x\right)$ is defined by

and $D_{0,k}\left(a,x\right)=2-k$.

In this talk, I will completely explain the permutation behaviour of the reversed Dickson polynomials of the $\left(k+1\right)$-th kind $D_{n,k}\left(a,x\right)$ when $a=0$, $n=p^l$, $n=p^l+1$, and $n=p^l+2$, where $l\ge 0$ is an integer. I will also explain the permutation behaviour of $D_{n,k}\left(1,x\right)$ when $n$ is a sum of odd prime powers. Moreover, I will present some algebraic and arithmetic properties of the reversed Dickson polynomials of the $\left(k+1\right)$-th kind.

In particular, I will explain the explicit evaluation of the sum ${\sum }_{a\in {\mathbb{𝔽}}_q}{D}_{n,k}\left(1,a\right)$ which provides a necessary condition for $D_{n,k}\left(1,x\right)$ to be a permutation polynomial of ${\mathbb{𝔽}}_q$. These results unify and generalize numerous recently discovered results on reversed Dickson polynomials over finite fields.