Polynomial functions over dual numbers of several variables

2020/02/03 by Amr Ali Abdulkader Al-Maktry, Al-Maktry, A. A. A.
Computer Science · Mathematics · #05A05 #06B10 #20B35 #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #Primary 13B25 #Secondary 12E10

paper · pdf · doi:10.48550/arxiv.2002.01304

openalex publication_date 2020/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k∈ ℕ∖\0\. For a commutative ring R, the ring of dual numbers of k variables over R is the quotient ring R[x₁,…,x_k]/ I , where I is the ideal generated by the set \x_ix_j| i,j=1,…,k\. This ring can be viewed as R[α₁,…,α_k] with α_i α_j=0, where α_i=x_i+I for i,j=1,…,k. We investigate the polynomial functions of R[α₁,…,α_k] whenever R is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on R[α₁,…,α_k] depending on the order of the pointwise stabilizer of the subring of constants R in the group of polynomial permutations of R[α₁,…,α_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[α₁,…,α_k] is a polynomial function if and only if a system of linear equations on R that depends on F has a solution.

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