Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Permutation polynomials and resolution of singularities over finite fields

Author: Da Qing Wan
Journal: Proc. Amer. Math. Soc. 110 (1990), 303-309
MSC: Primary 11T06
MathSciNet review: 1031673
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A geometric approach is introduced to study permutation polynomials over a finite field. As an application, we prove that there are no permutation polynomials of degree $ 2l$ over a large finite field, where $ l$ is an odd prime. This proves that the Carlitz conjecture is true for $ n = 2l$. Previously, the conjecture was known to be true only for $ n \leq 16$.

References [Enhancements On Off] (What's this?)

  • [1] S. D. Cohen, Permutation polynomials and primitive permutation groups, University of Glasgow, Department of Mathematics, preprint series no. 89/41. MR 1129514 (92j:11145)
  • [2] L. E. Dickson, The analytic representation of substitutions on a prime power of letters with a discussion of the linear group, Ann. of Math. 11 (1897), 65-120, 161-183.
  • [3] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, no. 52, Springer-Verlag, Berlin and New York, 1977, 28-30. MR 0463157 (57:3116)
  • [4] D. R. Hayes, A geometric approach to permutation polynomials over a finite field, Duke Math. J. 34 (1967), 293-305. MR 0209266 (35:168)
  • [5] R. Lidl and G. L. Mullen, When does a polynomial over a finite field permute the elements of the field?, Amer. Math. Monthly 95 (1988), 243-246. MR 1541277
  • [6] R. Lidl and H. Niderreiter, Finite fields, Encyclopedia Math. Appl., vol. 20, Addison-Wesley, Reading, MA, 1983.
  • [7] W. M. Schmidt, Equations over finite fields, Lecture Notes in Math., vol. 536, Springer-Verlag, Heidelberg, 1976. MR 0429733 (55:2744)
  • [8] B. Segre, Arithmetische Eigenschaften von Galois-Raumen, I, Math. Ann. 154 (1964), 195-256. MR 0165415 (29:2697)
  • [9] Daqing Wan, On a conjecture of Carlitz, J. Austral. Math. Soc. (Ser. A) 43 (1987), 375-384. MR 904396 (89c:11178)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11T06

Retrieve articles in all journals with MSC: 11T06

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society