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Polynomial functions on rings of dual numbers over residue class rings of the integers

  • Hasan Al-Ezeh , Amr Ali Al-Maktry EMAIL logo and Sophie Frisch
Published/Copyright: October 4, 2021
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Abstract

The ring of dual numbers over a ring R is R[α] = R[x]/(x2), where α denotes x + (x2). For any finite commutative ring R, we characterize null polynomials and permutation polynomials on R[α] in terms of the functions induced by their coordinate polynomials (f1, f2R[x], where f = f1 + αf2) and their formal derivatives on R.

We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on ℤpn[α] for np (p prime).


This work was supported by the Austrian Science Fund FWF projects P 27816-N26 and P 30934-N35.


Funding statement: The second and third authors wish to dedicate this paper to the memory of Prof. Al-Ezeh, who died while the paper was under review

  1. (Communicated by István Gaál)

Acknowledgement

The authors would like to thank Irena Swanson for valuable suggestions and comments on earlier versions of the manuscript.

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Received: 2020-02-22
Accepted: 2020-12-30
Published Online: 2021-10-04
Published in Print: 2021-10-26

© 2021 Mathematical Institute Slovak Academy of Sciences

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