@article{Han2023, 
author = {Lin Han and Guangyan Zhu and Zongbing Lin},
title = {On the rationality of generating functions of certain hypersurfaces over finite fields},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {6},
pages = {13898-13906},
keywords = {generating function, rationality, hypersurface, finite field},
url = {https://www.sciopen.com/article/10.3934/math.2023711},
doi = {10.3934/math.2023711},
abstract = {Let    a  ,  n be positive integers and let    p be a prime number. Let              F        q   be the finite field with    q  =      p    a   elements. Let    {      a    i        }          i      =      1        ∞   be an arbitrary given infinite sequence of elements in              F        q   and        a    1    ≠  0. For each positive integer    i, let    {      d          i      +      j      ,      i            }          j      =      0        ∞   be an arbitrary given sequence of positive integers with        d          i      i       coprime to    q  −  1. For each integer    n  ≥  1, let        N    n  ,                      N        ¯              n   and                      N        ~              n   denote the number of              F        q  -rational points of the hypersurfaces defined by the following three equations:         a    1        x    1    +  ⋯  +      a    n        x    n    =  b  ,         x    1    2    +  ⋯  +      x    n    2    =  band         a    1        x    1                  d                  11                      +      a    2        x    1                  d                  21                          x    2                  d                  22                      +  ⋯  +      a    n        x    1                  d                  n          1                          x    2                  d                  n          2                      ⋯      x    n                  d                  n          n                      =  b  ,respectively. In this paper, we show that the generating function        ∑          n      =      1              ∞            N    n        t    n   is a rational function in    t. Moreover, we show that if    p is an odd prime, then the generating functions        ∑          n      =      1              ∞                          N        ¯              n        t    n   and        ∑          n      =      1              ∞                          N        ~              n        t    n   are both rational functions in    t. Moreover, we present the explicit rational expressions of        ∑          n      =      1              ∞            N    n        t    n  ,        ∑          n      =      1              ∞                          N        ¯              n        t    n   and        ∑          n      =      1              ∞                          N        ~              n        t    n  , respectively.}
}