AIMS Mathematics, 2020, 5(4): 2979-2991. doi: 10.3934/math.2020192

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On the number of solutions of two-variable diagonal quartic equations over finite fields

1 School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, P. R. China
2 Mathematical College, Sichuan University, Chengdu 610064, P. R. China
3 Nanyang Normal University, Nanyang 473061, P. R. China

Let $p$ be a odd prime number and let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements. In this paper, by using the Gauss sum and Jacobi sum, we give an explicit formula for the number $N(x_1^4+x_2^4=c)$ of solutions of the following two-variable diagonal quartic equations over $\mathbb{F}_q$: $x_1^4+x_2^4=c$ with $c\in\mathbb{F}_q^*$. From this result, one can deduce that $N(x_1^4+x_2^4=c)=q+O(q^{\frac{1}{2}}).$
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1. A. Adolphson and S. Sperber, p-Adic estimates for exponential sums and the theorem of ChevalleyWarning, Ann. Sci.'Ecole Norm. Sup., 20 (1987), 545-556.    

2. A. Adolphson and S. Sperber, p-Adic estimates for exponential sums. In: F.Baldassarri, S. Bosch, B. Dwork (eds) p-adic Analysis. Lecture Notes in Mathematics, Springer, Berlin, 1990.

3. J. Ax, Zeros of polynomials over finite fields, Amer. J. Math., 86 (1964), 255-261.    

4. B. Berndt, R. Evans, K. Williams, Gauss and Jacobi Sums, Wiley-Interscience, New York, 1998.

5. L. Carlitz, The numbers of solutions of a particular equation in a finite field, Publ. Math. Debrecen, 4 (1956), 379-383.

6. S. Chowla, J. Cowles and M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory, 9 (1977), 502-506.    

7. S. F. Hong, Newton polygons of L-functions associtated with exponential sums of polynomials of degree four over finite fields, Finite Fields Th. App., 7 (2001), 205-237.    

8. S. F. Hong, Newton polygons of L-functions associtated with exponential sums of polynomials of degree six over finite fields, J. Number Theory, 97 (2002), 368-396.    

9. S. F. Hong, L-functions of twisted diagonal exponential sums over finite fields, Proc. Amer. Soc., 135 (2007), 3099-3108.    

10. S. F. Hong, J. R. Zhao and W. Zhao, The universal Kummer congruences, J. Aust. Math. Soc., 94 (2013), 106-132.    

11. S. N. Hu, S. F. Hong and W. Zhao, The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory, 156 (2015), 135-153.    

12. R. Lidl, H. Niederreiter, Finite Fields, second ed., Cambridge University Press, Cambridge, 1997.

13. G. Myerson, On the number of zeros of diagonal cubic forms, J. Number Theory, 11 (1979), 95-99.    

14. J. Wolfmann, The number of solutions of certain diagonal equations over finite fields, J. Number Theory, 42 (1992), 247-257.    

15. A. Weil, On some exponential sums, Proc. Natu. Acad. Sci., 34 (1948), 204-207.    

16. W. P. Zhang and J. Y. Hu, The number of solutions of the diagonal cubic congruence equation mod p, Math. Rep. (Bucur.), 20 (2018), 73-80.

17. J. Y. Zhao, S. F. Hong and C. X. Zhu, The number of rational points of certain quartic diagonal hypersurfaces over finite fields, AIMS Math., 5 (2020), 2710-2731.    

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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