Polynomial functions over dual numbers of several variables

Abstract

Let $k\in \mathbb{N}\setminus\{0\}$. For a commutative ring $R$, the ring of dual numbers of $k$ variables over $R$ is the quotient ring $R[x_1,\ldots,x_k]/ I $, where $I$ is the ideal generated by the set $\{x_ix_j\mid i,j=1,\ldots,k\}$. This ring can be viewed as $R[\alpha_1,\ldots,\alpha_k]$ with $\alpha_i \alpha_j=0$, where $\alpha_i=x_i+I$ for $i,j=1,\ldots,k$. We investigate the polynomial functions of $R[\alpha_1,\ldots,\alpha_k]$ whenever $R$ is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on $R[\alpha_1,\ldots,\alpha_k]$ depending on the order of the pointwise stabilizer of the subring of constants $R$ in the group of polynomial permutations of $R[\alpha_1,\ldots,\alpha_k]$. 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 $R[\alpha_1,\ldots,\alpha_k]$ is a polynomial function if and only if a system of linear equations on $R$ that depends on $F$ has a solution.