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

DOI:
https://doi.org/10.1090/S0002-9939-1990-1031673-4

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 over a large finite field, where is an odd prime. This proves that the Carlitz conjecture is true for . Previously, the conjecture was known to be true only for .

**[1]**Stephen D. Cohen,*Permutation polynomials and primitive permutation groups*, Arch. Math. (Basel)**57**(1991), no. 5, 417–423. MR**1129514**, https://doi.org/10.1007/BF01246737**[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]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[4]**D. R. Hayes,*A geometric approach to permutation polynomials over a finite field*, Duke Math. J.**34**(1967), 293–305. MR**0209266****[5]**Rudolf Lidl and Gary L. Mullen,*Unsolved Problems: When Does a Polynomial Over a Finite Field Permute the Elements of the Field?*, Amer. Math. Monthly**95**(1988), no. 3, 243–246. MR**1541277**, https://doi.org/10.2307/2323626**[6]**R. Lidl and H. Niderreiter,*Finite fields*, Encyclopedia Math. Appl., vol. 20, Addison-Wesley, Reading, MA, 1983.**[7]**Wolfgang M. Schmidt,*Equations over finite fields. An elementary approach*, Lecture Notes in Mathematics, Vol. 536, Springer-Verlag, Berlin-New York, 1976. MR**0429733****[8]**Beniamino Segre,*Arithmetische Eigenschaften von Galois-Räumen. I*, Math. Ann.**154**(1964), 195–256 (German). MR**0165415**, https://doi.org/10.1007/BF01362097**[9]**Da Qing Wan,*On a conjecture of Carlitz*, J. Austral. Math. Soc. Ser. A**43**(1987), no. 3, 375–384. MR**904396**

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

Retrieve articles in all journals with MSC: 11T06

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1031673-4

Article copyright:
© Copyright 1990
American Mathematical Society