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

TL;DR

This paper characterizes null and permutation polynomials over rings of dual numbers built on finite residue class rings of integers, providing explicit formulas for their counts in specific cases.

Contribution

It offers a complete characterization of null and permutation polynomials on dual number rings over finite rings and derives explicit formulas for their enumeration.

Findings

Characterization of null and permutation polynomials in dual number rings.

Explicit formulas for polynomial functions and permutations on $Z_{p^n}[alpha]$ for certain $n$.

Formulas relate polynomial properties to functions and derivatives over the base ring.

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).