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Factorization formulae on counting zeros of diagonal equations over finite fields


Authors: Wei Cao and Qi Sun
Journal: Proc. Amer. Math. Soc. 135 (2007), 1283-1291
MSC (2000): Primary 11T24, 11T06; Secondary 11D72
DOI: https://doi.org/10.1090/S0002-9939-06-08622-9
Published electronically: November 14, 2006
MathSciNet review: 2276636
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Abstract: Let $ N$ be the number of solutions $ (u_1,\ldots,u_n)$ of the equation $ a_1u_1^{d_1}+\cdots+a_nu_n^{d_n}=0$ over the finite field $ F_q$, and let $ I$ be the number of solutions of the equation $ \sum_{i=1}^nx_i/d_i\equiv 0\pmod{1}, 1\leqslant x_i\leqslant d_i-1$. If $ I>0$, let $ L$ be the least integer represented by $ \sum_{i=1}^nx_i/d_i, 1\leqslant x_i\leqslant d_i-1$. $ I$ and $ L$ play important roles in estimating $ N$. Based on a partition of $ \{d_1,\dots,d_n\}$, we obtain the factorizations of $ I, L$ and $ N$, respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for $ N$ in some special cases.


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  • 1. J. Ax, Zeros of polynomials over finite fields, Amer. J. Math. 86 (1964) 255-261. MR 0160775 (28:3986)
  • 2. B.C. Berndt, R.J. Evans, K.S. Williams, Gauss and Jacobi Sums, C.M.S. Series Monographs and Advanced Texts 21, Wiley-Interscience, New York, 1998. MR 1625181 (99d:11092)
  • 3. L.K. Hua, H.S. Vandiver, Characters over certain types of rings with applications to the theory of equations in a finite field, Proc. Nat. Acad. Sci. U.S.A. 35 (1949) 94-99. MR 0028895 (10:515c)
  • 4. K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1982.MR 0661047 (83g:12001)
  • 5. J.R. Joly, Équations et vari $ \rm\acute{e}$t $ \rm\acute{e}$s alg $ \rm\acute{e}$briques sur un corps fini, Enseign. Math. 19 (1973) 1-117. MR 0327723 (48:6065)
  • 6. R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, Vol. 20, Addison-Wesley, Reading, MA, 1983.MR 0746963 (86c:11106)
  • 7. C. Small, Diagonal equations over large finite fields, Canad. J. Math. 36 (1984) 249-262. MR 0749983 (86e:11094)
  • 8. Q. Sun, D. Wan, On the solvability of the equation $ \sum\nolimits_{i=1}^nx_i/d_i\equiv0\pmod{1}$ and its application, Proc. Amer. Math. Soc. 100 (1987) 220-224. MR 0884454 (88e:11015)
  • 9. Q. Sun, D. Wan, On the diophantine equation $ \sum\nolimits_{i=1}^nx_i/d_i\equiv0\pmod{1}$, Proc. Amer. Math. Soc. 112 (1991) 25-29. MR 1047008 (91h:11017)
  • 10. Q. Sun, D. Wan, D. Ma, On the Diophantine equation $ \sum\nolimits_{i=1}^nx_i/d_i\equiv0\pmod{1}$ and its applications, Chinese Ann. of Math., Ser. B 7 (1986) 232-236. MR 0858601 (87j:11023)
  • 11. Q. Sun, On diagonal equations over finite fields, Finite Fields Appl. 3 (1997) 175-179. MR 1444703 (97m:11050)
  • 12. Z.W. Sun, Exact $ m$-covers and the linear form $ \sum\nolimits_{s=1}^kx_s/n_s$, Acta Arith. 81 (1997) 175-198. MR 1456240 (98h:11019)
  • 13. D. Wan, Zeros of diagnol equations over finite fields, Proc. Amer. Math. Soc. 103 (1988) 1049-1052. MR 0954981 (89i:11138)
  • 14. A. Weil, Number of solutions of equations in a finite field, Bull. Amer. Math. Soc. 55 (1949) 497-508. MR 0029393 (10:592e)
  • 15. J. Wolfmann, New results on diagonal equations over finite fields from cyclic codes, Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 168 (1994) 387-395. MR 1291445 (95k:11161)

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Additional Information

Wei Cao
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China
Email: caowei433100@vip.sina.com

Qi Sun
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-06-08622-9
Keywords: Jacobi sum, Gauss sum, diagonal equation, finite fields
Received by editor(s): July 19, 2005
Received by editor(s) in revised form: December 21, 2005
Published electronically: November 14, 2006
Additional Notes: This work was partially supported by the National Natural Science Foundation of China, Grant #10128103.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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