Polynomial functions on rings of dual numbers over residue class rings of the integers

Abstract

The ring of dual numbers over a ring $R$ is $R[\alpha] = R[x]/(x^2)$, where $\alpha$ denotes $x+(x^2)$. For any finite commutative ring $R$, we characterize null polynomials and permutation polynomials on $R[\alpha]$ in terms of the functions induced by their coordinate polynomials ($f_1,f_2\in R[x]$, where $f=f_1+\alpha f_2$) and their formal derivatives on $R$. We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on $\mathbb{Z}_{p^n}[\alpha]$ for $n\le p$ ($p$ prime).