Polynomial functions on rings of dual numbers over residue class rings of the integers
Hasan Al-Ezeh, Amr Ali Abdulkader Al-Maktry, Sophie Frisch · Mathematica Slovaca · 2021
Abstract The ring of dual numbers over a ringRisR[α] =R[x]/(x2), whereαdenotesx+ (x2). For any finite commutative ringR, we characterize null polynomials and permutation polynomials onR[α] in terms of the functions induced by their coordinate polynomials (f1,f2∈R[x], wheref=f1+αf2) and their formal derivatives onR. We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on ℤpn[α] forn≤p(pprime).