@article{Hu2024, 
author = {Shuangnian Hu and Yanyan Li and Rongquan Feng},
title = {Counting rational points of quartic diagonal hypersurfaces over finite fields},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {1},
pages = {2167-2180},
keywords = {finite field, rational point, diagonal equation, Jacobi sum},
url = {https://www.sciopen.com/article/10.3934/math.2024108},
doi = {10.3934/math.2024108},
abstract = {Let              F        q   be the finite field of order    q where    q  =      p          k      ,    k is a positive integer and    p is an odd prime. Let              F        q    ∗   represent the nonzero elements of              F              q      . For    f  (      x    1    ,  ⋯  ,      x    n    )  ∈            F        q    [      x    1    ,  ⋯  ,      x    n    ], we used    N      (    f  (      x    1    ,  ⋯  ,      x    n    )  =  0      )   to denote the number of              F        q  -rational points of the affine hypersurface    f  (      x    1    ,  ⋯  ,      x    n    )  =  0. In 2020, Zhao et al. obtained the explicit formulae for    N  (      x    1    4    +      x    2    4    =  c  ),    N  (      x    1    4    +      x    2    4    +      x    3    4    =  c  ) and    N  (      x    1    4    +      x    2    4    +      x    3    4    +      x    4    4    =  c  ) over              F        q  , with    c  ∈            F        q    ∗  . In this paper, by using Jacobi sums and an analog of the Hasse-Davenport theorem, we arrived at explicit formulae for    N  (      a    1        x    1    4    +      a    2        x    2    4    =  c  ) and    N  (      b    1        x    1    4    +      b    2        x    2    4    +      b    3        x    3    4    =  c  ) with        a    i    ,      b    j    ∈            F        q    ∗    (  1  ≤  i  ≤  2  ,  1  ≤  j  ≤  3  ) and    c  ∈            F        q  . Furthermore, by using the reduction formula for Jacobi sums, the number of rational points of the quartic diagonal hypersurface        a    1        x    1    4    +      a    2        x    2    4    +  ⋯  +      a    n        x    n    4    =  c of    n  ≥  4 variables with        a    i    ∈            F        q    ∗      (  1  ≤  i  ≤  n  ),    c  ∈            F        q   and    p  ≡  1  (            m      o      d          4  ), can also be deduced. These extended and improved earlier results.}
}