The recurrence formula for the number of solutions of a equation in finite field

Yanbo Song; Yanbo Song

doi:10.3934/math.2021119

AIMS Mathematics

2021, Volume 6, Issue 2: 1954-1964. doi: 10.3934/math.2021119

Previous Article Next Article

Research article

The recurrence formula for the number of solutions of a equation in finite field

Yanbo Song ^,

School of mathematics and statistics, Xi'an Jiaotong university, Xianning west Road, Xi'an, China

Received: 29 August 2020 Accepted: 02 December 2020 Published: 07 December 2020
MSC : 11D45, 11D72

The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.
- recurrence formula,
- finite field,
- analytic methods,
- characters,
- Guass sum
Citation: Yanbo Song. The recurrence formula for the number of solutions of a equation in finite field[J]. AIMS Mathematics, 2021, 6(2): 1954-1964. doi: 10.3934/math.2021119

Related Papers:

Abstract

The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.

References

[1]	T. M. Apostol, Introduction to analytic number theory, New York: Springer-Verlag, 1976.
[2]	J. Ax, Zeros of polynomials over finite fields, Amer. J. Math., 86 (1964), 255–261. doi: 10.2307/2373163
[3]	B. C. Berndt, R. J. Evans, The determination of Gauss sums, B. Am. Math. Soc., 5 (1987), 107–130.
[4]	J. Bourgain, M. C. Chang, A Gauss sum estimate in arbitrary finite fields, C. R. Math., 342 (2006), 643–646. doi: 10.1016/j.crma.2006.01.022
[5]	W. Cao, A special degree reduction of polynomial over finite fields with applications, Int. J. Number Theory, 7 (2011), 1093–1102. doi: 10.1142/S1793042111004277
[6]	W. Cao, Q. Sun, On a class of equations with special degrees over finite fields, Acta Arith., 130 (2007), 195–202. doi: 10.4064/aa130-2-8
[7]	S. Chowla, J. Cowles, M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory, 9 (1977), 502–506. doi: 10.1016/0022-314X(77)90010-5
[8]	M. Z. Garaev, On the gauss trigonometric sum, Math. Notes, 68 (2000), 154–158. doi: 10.1007/BF02675340
[9]	S. Hu, J. Zhao, The number of rational points of a family of algebraic varieties over finite fields, Algebra Colloq., 24 (2017), 705–720. doi: 10.1142/S1005386717000475
[10]	S. Hu, S. Hong, W. Zhao, The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory, 156 (2015), 135–153. doi: 10.1016/j.jnt.2015.04.006
[11]	H. Huang, W. Gao, W. Cao, Remarks on the number of rational points on a class of hypersurfaces over finite fields, Algebra Colloq., 25 (2018), 533–540. doi: 10.1142/S1005386718000366
[12]	G. Myerson, On the number of zeros of diagonal cubic forms, J. Number Theory, 11 (1979), 95–99. doi: 10.1016/0022-314X(79)90023-4
[13]	Kh. M. Saliba, V. N. Chubarikov, A generalizition of the Gauss sum, Moscow Univ. Math. Bull., 64 (2009), 92–94.
[14]	W. M. Schmidt, Equations over finite fields: an elementary approach, New York: Springer-Verlag, 1976.
[15]	A. Weil, On some exponential sums, P. Natl. Acad. Sci. USA, 34, (1948), 203–210.
[16]	W. Zhang, J. Zhang, Some character sums of the polynomials, JP Journal of Algebra, Number Theory and Applications, 48 (2020), 37–48. doi: 10.17654/NT048010037
[17]	J. Zhao, S. Hong, C. Zhu, The number of rational points of certain quartic diagonal hypersurfaces over finite fields, AIMS Mathematics, 5 (2020), 2710–2731. doi: 10.3934/math.2020175

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)