Polynomial Functions over Dual Numbers of Several Variables

Let k be a positive integer. For a commutative ring R, the ring of dual numbers of k variables over R is the quotient ring R[x1,...,xk]/I, where I is the ideal generated by the set {xixj|i,j {1,...,k}}. This ring can be viewed as R[α1,...,αk] with αiαj = 0, where αi = xi + I for 1 ≤ i,j ≤ k. We investigate the polynomial functions of R[α1,...,αk] whenever R is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on R[α1,...,αk] depending on the order of the pointwise stabilizer of the subring of constants R in the group of polynomial permutations of R[α1,...,αk]. Further, we show that the stabilizer group of R is independent of the number of variables k. Moreover, we prove that a function F on R[α1,...,αk] is a polynomial function if and only if a system of linear equations on R that depends on F has a solution.