## Abstract

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 language | English |
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Pages (from-to) | 521 - 537 |

Number of pages | 17 |

Journal | Applicable Algebra in Engineering, Communication and Computing |

Volume | 34 |

Issue number | 3 |

Early online date | 29 May 2021 |

DOIs | |

Publication status | Published - May 2023 |

## Keywords

- 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