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## Proceedings of the Edinburgh Mathematical Society (Series 2)

### Research Article

### The number of permutation polynomials of the form *f*(*x*) *cx* over a finite field

### Daqing Wan^{a1}*, Gary L. Mullen^{a2}** and Peter Jau-Shyong Shiue^{a3}*

- Proceedings of the Edinburgh Mathematical Society (Series 2) / Volume 38 / Issue 01 / February 1995, pp 133-149
- Copyright © Edinburgh Mathematical Society 1995
- DOI: http://dx.doi.org/10.1017/S001309150000626X (About DOI), Published online: 20 January 2009

^{a1 }Department of Mathematical Sciences University of Nevada, Las Vegas, NV 89154 USA e-mail: dwan@nevada.edu

^{a2 }Department of Mathematics The Pennsylvania State University University Park, PA 16802 USA e-mail: mullen@math.psu.edu

^{a3 }Department of Mathematical Sciences University of Nevada. Las Vegas, NV 89154 USA e-mail: shiue@nevada.edu

Abstract

Let *F*^{q} be the finite field of *q* elements. Let *f*(*x*) be a polynomial of degree *d* over *F*^{q} and let *r* be the least non-negative residue of q-1 modulo d. Under a mild assumption, we show that there are at most *r* values of *c**F*^{q}, such that *f*(*x*) + *cx* is a permutation polynomial over *F*^{q}. This indicates that the number of permutation polynomials of the form *f*(*x*) +*cx* depends on the residue class of *q*–1 modulo *d*.

As an application we apply our results to the construction of various maximal sets of mutually orthogonal latin squares. In particular for odd *q* = *p ^{n}* if τ(

(Received June 17 1993)

Footnotes

* These authors would like to thank UNLV for partial support by the University Research Grants and Fellowship Committee. The first author would also like to thank the Institute for Advanced Study for its hospitality, and the National Science Foundation and the UNLV Faculty Development Leave Committee for partial support.

** This author would like to thank the National Security Agency for partial support under grant agreement #MDA904-92-H-3044.