Algebraic Structure of Permutational Polynomials over $\mathbb{F}_{q^n}$ \uppercase\expandafter{\romannumeral2}
Pingzhi Yuan, Xuan Pang, Danyao Wu
TL;DR
The paper addresses the problem of clarifying how the vector space $\\mathbb{F}_{q}^n$ and the finite field $\\mathbb{F}_{q^n}$ relate algebraically, and develops a generalized framework that expresses a permutation polynomial $f$ on $\\mathbb{F}_{q^n}$ as $F(x)=\eta\circ g\circ\rho\circ f(x)$ with $\\rho: \\mathbb{F}_{q^n} \to \\mathbb{F}_{q}^n$ and $\\eta: \\mathbb{F}_{q}^n \to \\mathbb{F}_{q^n}$ being $\\mathbb{F}_q$-isomorphisms and $g: \\mathbb{F}_{q}^n \to \\mathbb{F}_{q}^n$ a PP on $\\mathbb{F}_{q}^n$. This framework yields bidirectional correspondences between PPs on the two spaces (via the corollaries) and provides a constructive route to new PPs. The authors then apply the theory to construct several classes of permutation polynomials over $\\mathbb{F}_{q^2}$ of the form $x^r h(x^{q-1})$, deriving explicit necessary and sufficient conditions (in terms of gcds and linear independence of basis elements) and detailing numerous cases for $q$ modulo $3$. These results unify and extend known Zieve/Ding–Zieve-type constructions, offering a systematic algebraic lens and potential pathways to additional PPs and related objects such as planar and APN functions.
Abstract
It is well known that there exists a significant equivalence between the vector space $\mathbb{F}_{q}^n$ and the finite fields $\mathbb{F}_{q^n}$, and many scholars often view them as the same in most contexts. However, the precise connections between them still remain mysterious. In this paper, we first show their connections from an algebraic perspective, and then propose a more general algebraic framework theorem. Furthermore, as an application of this generalized algebraic structure, we give some classes of permutation polynomials over $\mathbb{F}_{q^2}$.