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.
Categories: math.AC