F_(p^(2r))上一类指数和的值

The Values of a Class of Exponential Sums over F_(p^(2r))

设p为奇质数,q=p^(n)(n∈Z^(+)),f(x)∈F_(q)[x].Wan(Journal of Systems Science and Complexity,2021,34(4):498-510.)研究了指数和S_(q)(f(x))的代数次数及周期性.特别地,陈超等(四川大学学报(自然科学版),2020,57(6):1029-1032.)证明了p≡-1(mod d)且q=p^(2)时,S_(q)(x^(d))=-p或(d-1)p.基于上述结果,讨论p≡-1(mod d)且q=p^(2r)(r∈Z^(+))时S_(q)(x^(d))的具体取值,进而利用初等的方法和技巧,得到deg S_(q)(x^(d))=1时,S_(q)(x^(d))的全部可能取值.

Let p be an odd prime,q=pn(n∈Z^(+)),f(x)∈F_(q)[x].Recently Professor Daqing Wan(Journal of Systems Science and Complexity,2021,34(4):498-510.)studied the algebraic degrees and the periods of the exponential sums S_(q)(f(x))over the finite field F_(q).In addition,Chao Chen,et al(Journal of Sichuan University(Natural Science Edition),2020,57(6):1029-1032.)proved that S_(q)(x^(d))has only two possible values for the case with both p≡-1(mod d)and q=p^(2).In this paper,based on Wan and Chen et al’s results,we study the value of S_(q)(x^(d))for the case with p≡-1(mod d)and deg S_(q)(x^(d))=1,and then obtain the all possible values of S_(q)(x^(d))in this case.