On the group of unit-valued polynomial functions

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 ${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[x]/(x^2))\cong \mathcal{P}({\mathbb{F}}_q){⋉}_{\theta }\mathcal{F}({\mathbb{F}}_q{)}^{\times }$. Furthermore, we count unit-valued polynomial functions $(\mathrm{mod}{p}^n)$ and obtain canonical representations for these functions.

Cited by in corpus (1)

• The Ring of Polyfunctions over $\mathbb{Z}/n\mathbb{Z}$