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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Solution to a problem of S. Payne
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by Xiang-dong Hou PDF
Proc. Amer. Math. Soc. 132 (2004), 1-6 Request permission

Abstract:

A problem posed by S. Payne calls for determination of all linearized polynomials $f(x)\in \mathbb {F}_{2^n}[x]$ such that $f(x)$ and $f(x)/x$ are permutations of $\mathbb {F}_{2^n}$ and $\mathbb {F}_{2^n}^*$ respectively. We show that such polynomials are exactly of the form $f(x)=ax^{2^k}$ with $a\in \mathbb {F}_{2^n}^*$ and $(k,n)=1$. In fact, we solve a $q$-ary version of Payne’s problem.
References
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Additional Information
  • Xiang-dong Hou
  • Affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435
  • Address at time of publication: Department of Mathematics, University of South Florida, Tampa, Florida 33620
  • Email: xhou@euler.math.wright.edu
  • Received by editor(s): July 29, 2002
  • Published electronically: August 13, 2003
  • Additional Notes: This research was supported by NSA grant MDA 904-02-1-0080
  • Communicated by: David E. Rohrlich
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 1-6
  • MSC (2000): Primary 11T06; Secondary 51E20
  • DOI: https://doi.org/10.1090/S0002-9939-03-07240-X
  • MathSciNet review: 2021242