Contributions to the famiily of reversed Dickson polynomials

Abstract: Let $p$ be a prime, $q$ a power of $p$, and $\mathbb{F}_q$ the finite field with $q$ elements. A polynomial $f\in \mathbb{F}_q[\tt X]$ is called a permutation polynomial of $\mathbb{F}_q$ if the associated mapping $\tt X\mapsto f(\tt X)$ from $\mathbb{F}_q$ to $\mathbb{F}_q$ is a permutation of $\mathbb{F}_q$. Permutation polynomials have gained widespread attention due to their applications in cryptography, coding theory, and combinatorics. The $n$th reversed Dickson polynomial is given by the explicit expression $$ D_n(a,\tt X)=\sum_{i=0}^{\lfloor n/2\rfloor}\,\frac{n}{n-i}\,\binom{n-i}{i}\,a^{n-2i}\,(-\tt X)^i $$ where $a\in \mathbb{F}_q$ is a parameter. Reversed Dickson polynomials have played an important role in the area of permutation polynomials since their introduction in 2009.

A self-reciprocal polynomial is a polynomial whose coefficients form a palindrome. Self-reciprocal polynomials have important applications in coding theory. In this talk, I will first speak about my contribution to the areas of permutation polynomials over finite fields and self-reciprocal polynomials via reversed Dickson polynomials. I will also speak about a recent REU project conducted with my students at College of the Holy Cross on reversed Dickson permutation polynomials. Moreover, I will present a list of research projects for students.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)