On the group of unit-valued polynomial functions

TL;DR

This paper investigates the structure of polynomial functions over finite rings, especially their units and permutations, providing algebraic decompositions and counting formulas for functions modulo prime powers.

Contribution

It introduces a semidirect product structure for polynomial permutation groups over certain finite rings and counts unit-valued polynomial functions modulo prime powers.

Findings

The group of polynomial permutations on $R[x]/(x^2)$ embeds into a semidirect product involving units and permutations.

Explicit isomorphisms are established for polynomial permutation groups over finite fields.

Counting formulas for unit-valued polynomial functions modulo $p^n$ are derived.

Abstract

Let $R$ be a finite commutative ring with $1\ne 0$. The set $\mathcal{F}(R)$ of polynomial functions on $R$ is a finite commutative ring with pointwise operations. Its group of units $\mathcal{F}(R)^\times$ is just the set of all unit-valued polynomial functions, that is the set of polynomial functions which map $R$ into its group of units. We show that $\mathcal{P}_R(R[x]/(x^2))$ the group of polynomial permutations on the ring $R[x]/(x^2)$, consisting of permutations represented by polynomials over $R$, is embedded in a semidirect product of $\mathcal{F}(R)^\times$ by $\mathcal{P}(R)$ the group of polynomial permutations on $R$. In particular, when $R=\mathbb{F}_q$, we prove that $\mathcal{P}_{\mathbb{F}_q}(\mathbb{F}_q[x]/(x^2))\cong \mathcal{P}(\mathbb{F}_q) \ltimes_\theta \mathcal{F}(\mathbb{F}_q)^\times$. Furthermore, we count unit-valued polynomial functions $\pmod{p^n}$ and obtain canonical representations for these functions.