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