Polynomial functions over dual numbers of several variables

Abstract

Let [Formula: see text] be a positive integer. For a commutative ring [Formula: see text], the ring of dual numbers of [Formula: see text] variables over [Formula: see text] is the quotient ring [Formula: see text], where [Formula: see text] is the ideal generated by the set [Formula: see text]. This ring can be viewed as [Formula: see text] with [Formula: see text], where [Formula: see text] for [Formula: see text]. We investigate the polynomial functions of [Formula: see text] whenever [Formula: see text] is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on [Formula: see text] depending on the order of the pointwise stabilizer of the subring of constants [Formula: see text] in the group of polynomial permutations of [Formula: see text]. Further, we show that the stabilizer group of [Formula: see text] is independent of the number of variables [Formula: see text]. Moreover, we prove that a function [Formula: see text] on [Formula: see text] is a polynomial function if and only if a system of linear equations on [Formula: see text] that depends on [Formula: see text] has a solution.