@article{Chen2023, 
author = {Qinlong Chen and Wei Cao},
title = {The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {7},
pages = {4303-4312},
keywords = {finite field, polynomial, Jacobi sum, Gauss sum},
url = {https://www.sciopen.com/article/10.3934/era.2023219},
doi = {10.3934/era.2023219},
abstract = {Let    q be an even prime power and let              F              q       be the finite field of    q elements. Let    f be a nonzero polynomial over              F                      q        2             of the form    f  =      a          1            x          1                      m                  1                      +  ⋯  +      a          s            x          s                      m                  s                      +      y          1            y          2        +  ⋯  +      y          n      −      1            y          n        +      y          n      −      2      t      −      1              2        +  ⋯  +      y          n      −      3        2    +      y          n      −      1              2          +      b          t            y          n      −      2      t              2        +    ⋯  +      b          1            y          n      −      2              2        +      b          0            y          n              2      , where        a          i        ,      b    j    ∈            F                      q                  2                            ∗        ,        m    i    ≠  1  ,    (      m          i        ,      m          k        )  =  1  ,    i  ≠  k  ,        m          i            |    (  q  +  1  )  ,        m          i        ∈            Z              +        ,    2      |    n,    n  >  2,    0  ≤  t  ≤      n    2    −  2,                      T        r                                      F                                      q            2                                      /                              F                          2                      (      b          j        )  =  1 for    i  ,  k  =  1  ,  …  ,  s and    j  =  0  ,  1  ,  …  ,  t. For each    b  ∈            F                      q        2            , let        N                  q        2              (  f  =  b  ) denote the number of              F                      q        2            -rational points on the affine hypersurface    f  =  b. In this paper, we obtain the formula of        N                  q        2              (  f  =  b  ) by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields.}
}