Value sets of polynomials over finite fields

Authors:
Da Qing Wan, Peter Jau-Shyong Shiue and Ching Shyang Chen

Journal:
Proc. Amer. Math. Soc. **119** (1993), 711-717

MSC:
Primary 11T06; Secondary 11T55

DOI:
https://doi.org/10.1090/S0002-9939-1993-1155603-2

MathSciNet review:
1155603

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Abstract: Let be the finite field of elements, and let be the number of values taken by a polynomial over . We establish a lower bound and an upper bound of in terms of certain invariants of . These bounds improve and generalize some of the previously known bounds of . In particular, the classical Hermite-Dickson criterion is improved. Our bounds also give a new proof of a recent theorem of Evans, Greene, and Niederreiter. Finally, we give some examples which show that our bounds are sharp.

**[1]**L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Strauss,*Polynomials over finite fields with minimal value sets*, Mathematika**8**(1961), 121-130. MR**0139606 (25:3038)****[2]**W. S. Chou, J. Gomez-Calderon, and G. L. Mullen,*Value sets of Dickson polynomials over finite fields*, J. Number Theory**30**(1988), 334-344. MR**966096 (90e:11181)****[3]**R. J. Evans, J. Greene, and H. Niederreiter,*Linearized polynomials and permutation polynomials of finite fields*, Michigan Math. J. (to appear). MR**1182496 (93j:11080)****[4]**J. Gomez-Calderon,*A note on polynomials with minimal value set over finite fields*, Mathematika**35**(1988), 144-148. MR**962743 (90e:11183)****[5]**J. Gomez-Calderon and D. J. Madden,*Polynomials with small value sets over finite fields*, J. Number Theory**28**(1988), 167-188. MR**927658 (89d:11111)****[6]**V. A. Kurbatov and N. G. Starkov,*The analytic representation of permutations*, Sverdlovsk. Gos. Ped. Inst. Ucen. Zat.**31**(1965), 151-158. (Russian) MR**0215817 (35:6652)****[7]**R. Lidl and H. Niederreiter,*Finite fields*, Addison-Wesley, Reading, MA, 1983. MR**746963 (86c:11106)****[8]**W. H. Mills,*Polynomials with minimal value sets*, Pacific J. Math.**14**(1964), 225-241. MR**0159813 (28:3029)****[9]**G. L. Mullen,*Permutation polynomials over finite fields*, Proceedings of the International Conference on Finite Fields, Coding Theory and Advances in Communications and Computing, Lecture Notes in Pure and Appl. Math., vol. 141, Marcel Dekker, New York, 1992, pp. 131-151. MR**1199828 (94d:11097)****[10]**L. J. Rogers,*Note on functions proper to represent a substitution of a prime number of letters*, Messenger Math.**21**(1981), 44-47.**[11]**D. Wan,*A*-*adic lifting lemma and its applications to permutation polynomials*, Proceedings of the International Conference on Finite Fields, Coding Theory and Advances in Communications and Computing, Lecture Notes in Pure and Appl. Math., vol. 141, Marcel Dekker, New York, 1992, pp. 209-216. MR**1199834 (93m:11129)****[12]**D. Wan and R. Lidl,*Permutation polynomials of the form**and their group structure*, Monatsh Math.**112**(1991), 149-163. MR**1126814 (92g:11119)**

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1155603-2

Article copyright:
© Copyright 1993
American Mathematical Society