On the group of unit-valued polynomial functions

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Let R be a finite commutative ring. The set F(R) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units F(R) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on R[x] / (x 2) = R[α] , the ring of dual numbers over R, and show that the group P R(R[α]) , consisting of those polynomial permutations of R[α] represented by polynomials in R[x], is embedded in a semidirect product of F(R) × by the group P(R) of polynomial permutations on R. In particular, when R= F q, we prove that PFq(Fq[α])≅P(Fq)⋉θF(Fq)×. Furthermore, we count unit-valued polynomial functions on the ring of integers modulo p n and obtain canonical representations for these functions.

Original languageEnglish
Pages (from-to)521 - 537
Number of pages17
JournalApplicable Algebra in Engineering, Communication and Computing
Issue number3
Early online date29 May 2021
Publication statusPublished - May 2023


  • Dual numbers
  • Finite commutative rings
  • Permutation polynomials
  • Polynomial functions
  • Polynomial mappings
  • Polynomial permutations
  • Semidirect product
  • Unit-valued polynomial functions

ASJC Scopus subject areas

  • Applied Mathematics
  • Algebra and Number Theory

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