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On a Conjecture of Carlitz

Published online by Cambridge University Press:  09 April 2009

Wan Daqing
Wan Daqing
Affiliation:
Department of MathematicsThe University of WashingtonSeattle, Washington 98195, U.S.A.
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A conjecture of Carlitz on permutation polynomials is as follow: Given an even positive integer n, there is a constant Cn, such that if Fq is a finite field of odd order q with q > Cn, then there are no permutation polynomials of degree n over Fq. The conjecture is a well-known problem in this area. It is easily proved if n is a power of 2. The only other cases in which solutions have been published are n = 6 (Dickson [5]) and n = 10 (Hayes [7]); see Lidl [11], Lausch and Nobauer [9], and Lidl and Niederreiter [10] for remarks on this problem. In this paper, we prove that the Carlitz conjecture is true if n = 12 or n = 14, and give an equivalent version of the conjecture in terms of exceptional polynomials.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 3 , December 1987 , pp. 375 - 384
Copyright
Copyright © Australian Mathematical Society 1987

References

1.Bombieri, E. and Davenport, H., ‘On two problems of Mordell’, Amer. J. Math. 88 (1966), 61–70.CrossRefGoogle Scholar
2.Cohen, S. D., ‘The distribution of polynomials over finite fields’, Ada. Arith. 17 (1970), 255–271.CrossRefGoogle Scholar
3.Davenport, H. and Lewis, D. J., ‘Notes on congruences (II)’, Quart. J. Math. (2) 14 (1963), 51–60.CrossRefGoogle Scholar
4.Dickson, L. E., Linear groups with an exposition of the Galois field theory (Dover, New York, 1958).Google Scholar
5.Dickson, L. E., ‘The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group’, Ann. of Math. 11 (1897), 65–120, 161–183.CrossRefGoogle Scholar
6.Gwehenberger, G., Über die Darstellung von Permutationene durch Polynome und rationale Funktionen (Diss. TH Wien. 1970).Google Scholar
7.Hayes, D. R., ‘A geometric approach to permutation polynomials over a finite field’, Duke Math. J. 34 (1967), 293–305.CrossRefGoogle Scholar
8.Lang, S. and Weil, A., ‘Number of points of varietes in finite fields’, Amer. J. Math. 76 (1953), 819–827.CrossRefGoogle Scholar
9.Lausch, H. and Nöbauer, W., Algebra of polynomials (North-Holland, Amsterdam, 1973).Google Scholar
10.Lidl, R. and Niederreiter, H., Finite fields, (Addison-Wesley, Reading, Massachusetts, 1983).Google Scholar
11.Lidl, R., ‘Einige ungelöste Probleme bei endlichen Körpern’, Math. Balkanica 4 (1974), 409–414.Google Scholar
12.MacCluer, C. R., ‘On a conjecture of Davenport and Lewis concerning exceptional polynomials’, Acda. Arith. 12 (1967), 289–299.CrossRefGoogle Scholar
13.Tietäväinen, A., ‘On non-residues of polynomial’, Ann. Univ. Turku Ser. AI 94 (1966), 6 pp.Google Scholar
14.van der Waerden, B. L., Modern algebra (Frederick Ungar, New York, 1953).Google Scholar
15.Williams, K. S., ‘On exceptional polynomials’, Canad. Math. Bull. 11 (1968), 179–282.CrossRefGoogle Scholar