Polynomial functions on rings of dual numbers over residue class rings of the integers. (English) Zbl 1484.11213
Summary: The ring of dual numbers over a ring \(R\) is \(R[\alpha]=R[x]/(x^2)\), where \(\alpha\) denotes \(x+(x^2)\). For any finite commutative ring \(R\), we characterize null polynomials and permutation polynomials on \(R[\alpha]\) in terms of the functions induced by their coordinate polynomials \((f_1, f_2 \in R[x]\), where \(f=f_1+\alpha f_2)\) and their formal derivatives on \(R\).
We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on \(\mathbb{Z}_{p^n}[\alpha]\) for \(n\leq p(p\) prime).
We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on \(\mathbb{Z}_{p^n}[\alpha]\) for \(n\leq p(p\) prime).
MSC:
| 11T06 | Polynomials over finite fields |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 13M10 | Polynomials and finite commutative rings |
| 13B25 | Polynomials over commutative rings |
| 12E10 | Special polynomials in general fields |
| 05A05 | Permutations, words, matrices |
| 06B10 | Lattice ideals, congruence relations |
Keywords:
finite rings; finite commutative rings; dual numbers; polynomials; polynomial functions; polynomial mappings; polynomial permutations; permutation polynomials; null polynomialsReferences:
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