@article{Hu2023, 
author = {Shuangnian Hu and Rongquan Feng},
title = {The number of solutions of cubic diagonal equations over finite fields},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {3},
pages = {6375-6388},
keywords = {finite fields, rational points, diagonal equations, Jacobi sums},
url = {https://www.sciopen.com/article/10.3934/math.2023322},
doi = {10.3934/math.2023322},
abstract = {Let    p be a prime,    k be a positive integer,    q  =      p    k  , and              F        q   be the finite field with    q elements. Let              F        q    ∗   be the multiplicative group of              F              q      , that is              F        q    ∗    =            F              q        ∖  {  0  }. In this paper, explicit formulae for the numbers of solutions of cubic diagonal equations        a    1        x    1    3    +      a    2        x    2    3    =  c and        b    1        x    1    3    +      b    2        x    2    3    +      b    3        x    3    3    =  c over              F        q   are given, with        a    i    ,      b    j    ∈            F        q    ∗      (  1  ≤  i  ≤  2  ,  1  ≤  j  ≤  3  ),    c  ∈            F        q   and    p  ≡  1  (      m    o    d      3  ). Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the cubic diagonal equations        a    1        x    1    3    +      a    2        x    2    3    +  ⋯  +      a    s        x    s    3    =  c of    s  ≥  4 variables with        a    i    ∈            F        q    ∗      (  1  ≤  i  ≤  s  ),    c  ∈            F        q   and    p  ≡  1  (      m    o    d      3  ), can also be deduced.}
}