Polynomial Functions over Dual Numbers of Several Variables

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Abstract

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.

Original languageEnglish
Article number2350231
JournalJournal of Algebra and its Applications
Early online date13 Sep 2022
DOIs
Publication statusE-pub ahead of print - 13 Sep 2022

Keywords

  • dual numbers
  • Finite commutative rings
  • finite polynomial permutation groups
  • null polynomials
  • permutation polynomials
  • polynomial functions
  • polynomial permutations
  • polynomials

ASJC Scopus subject areas

  • Applied Mathematics
  • Algebra and Number Theory

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