The number of solutions of diagonal cubic equations over finite fields

被引：3
|
作者：
Ge, Wenxu [1 ]
Li, Weiping [2 ]
Wang, Tianze [1 ]
机构：
[1] North China Univ Water Resources & Elect Power, Sch Math & Stat, Zhengzhou 450046, Peoples R China
[2] Henan Univ Econ & Law, Sch Math & Informat Sci, Zhengzhou 450046, Peoples R China
来源：
FINITE FIELDS AND THEIR APPLICATIONS | 2022年 / 80卷
基金：
中国国家自然科学基金;
关键词：
Gauss sum; Jacobi sum; Generating function; Diagonal cubic equation; Exponential sum; ZEROS;
D O I：
10.1016/j.ffa.2022.102008
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
Let Fq be a finite field of q = pk elements. For any z & ISIN; Fq, let An(z) and Bn(z) denote the number of solutions of the equations x31 + x32 + & BULL; & BULL; & BULL; + x3n = z and x31 + x32 + & BULL; & BULL; & BULL; + x3n + zx3n+1 = 0 respectively. Recently, using the generator of Fq*, Hong and Zhu gave the generating functions E & INFIN; and E & INFIN;n=1 An(z)xn n=1 Bn(z)xn. In this paper, we give the generating functions E & INFIN;n=1 An(z)xn and E & INFIN;n=1 Bn(z)xn immediately by the coefficient z. Moreover, we gave the formulas of the number of solutions of equation a1x31 + a2x32 + a3x33 = 0 and our formulas are immediately determined by the coefficients a1, a2 and a3. These extend and improve earlier results. (c) 2022 Elsevier Inc. All rights reserved.
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页数：16
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