Permutation polynomials, projective polynomials, and bijections between $\mu_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$

TL;DR

This paper introduces generalized Möbius transformations and projective polynomials to establish bijections between roots of unity and projective geometries, leading to new permutation polynomials over finite fields.

Contribution

It develops a framework using arbitrary bases to construct bijections and permutation polynomials connecting roots of unity and projective spaces over finite fields.

Findings

Established generalized Möbius transformations as bijections.

Determined inverses of the introduced projective polynomials.

Constructed permutation polynomials of specific index over finite fields.

Abstract

Using arbitrary bases for the finite field $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, we obtain the generalized M\"obius transformations (GMTs), which are a class of bijections between the projective geometry $PG(n-1,q)$ and the set of roots of unity $\mu_{\frac{q^n-1}{q-1}}\subseteq\mathbb{F}_{q^n}$, where $n\geq 2$ is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of $\mathbb{F}_{q^n}^\ast,\mu_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$. Through this correspondence and the GMTs, we construct permutation polynomials of index $\frac{q^n-1}{q-1}$ over $\mathbb{F}_{q^n}$.