Permutation Polynomials with Carlitz Rank 2

TL;DR

This paper establishes a sharp lower bound on the weight of permutation polynomials with Carlitz rank 2 over finite fields, advancing understanding of their complexity.

Contribution

It improves the existing lower bound for the weight of permutation polynomials with Carlitz rank 2, providing a more precise measure of their complexity.

Findings

Established a sharp lower bound for polynomial weight

Enhanced previous bounds by Gomez-Prez et al.

Deepened understanding of permutation polynomial complexity

Abstract

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.