# Permutation Polynomials over Finite Fields (19frg263)

## Organizers

(New York City College of Technology)

Daniele Bartoli (Università degli Studi di Perugia)

Steven Wang (Carleton University)

## Description

The Banff International Research Station will host the "Permutation Polynomials over Finite Fields" workshop in Banff from June 9, 2019 to June 16, 2019.

This workshop is a satellite event of The $14^{\mathrm{th}}$ International Conference on Finite Fields and Their Applications to be held in Vancouver, June 3-7, 2019. The goal is to gather a group of eight researchers interested in permutation polynomials over finite fields to work on problems of significant importance that require synergetic efforts of experts in the area. A polynomial $f\in\mathbb{F}_q[x]$ is a permutation polynomial over $\mathbb F_q$ if the induced mapping $f\colon\mathbb F_q\to\mathbb F_q$ is a bijection. Permutation polynomials have been studied since the end of the $19^{\mathrm{th}}$ century. In past decades, the research has been intensified due to their connections with applications in Cryptography, Coding Theory, and Combinatorial Designs. A major problem is to find classes of permutation polynomials over finite fields. In 2009 Akbary, Ghioca, and Wang introduced the concept of index of a polynomial to study the distribution of permutation polynomials over finite fields. They showed that the density of permutation polynomials in the set of polynomials with prescribed index and exponents is higher when the index is smaller. Since then, many authors have explored the index of a polynomial in a variety of questions, not only about permutation polynomials. Very recently, Wang wrote a survey on the index approach where a long section is dedicated to classifying permutation polynomials from over 60 papers based on their indices. We note that many results in these papers do not mention the index of the polynomials involved explicitly. So this survey is an important and major step in trying to organize the many existing permutation polynomials. It comes to confirm that the index approach is very promising, and it provides a rich source of directions of research. This workshop will give a group of researchers the unique opportunity to interact, concentrate efforts, and focus on problems related to the classification of permutation polynomials based on their indices. We expect this gathering to be very productive and result in meaningful publications that will be valuable for the research community.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).