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PERMUTATION POLYNOMIALS OF DEGREE 8 OVER FINITE FIELDS OF ODD CHARACTERISTIC

Part of: Finite fields and commutative rings (number-theoretic aspects) Computational aspects of field theory and polynomials

Published online by Cambridge University Press:  09 July 2019

XIANG FAN Open the ORCID record for XIANG FAN [Opens in a new window]
XIANG FAN*
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China email fanx8@mail.sysu.edu.cn
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Abstract

We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

Keywords

permutation polynomialHermite’s criterionCarlitz conjectureSageMath

MSC classification

Primary: 11T06: Polynomials
Secondary: 12Y05: Computational aspects of field theory and polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 1 , February 2020 , pp. 40 - 55
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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Footnotes

This work was partially supported by the Natural Science Foundation of Guangdong Province (No. 2018A030310080). The author was also sponsored by the National Natural Science Foundation of China (No. 11801579).

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