The number of solutions of certain quartic diagonal equations over finite field
Let p be a prime,k be a positive integer and Fq be the finite field of q= pk elements.Let F*q be the multiplicative group of Fq,that is,F*q=Fq\{0}.For a polynomial f(x1,…,xn)over Fq,use N(f(x1,…,xn)=0)to denote the number of solutions of f(x1,x2,…,xn)=0 over Fq.In 1981,Myerson gave a formula for N(x41 +⋯+ x4n=0).Recently,Zhao and coworkers obtained the explicit formulas for N(x41 + x42 = c),N(x41 + x42 + x43 = c)and N(x41 + x42 + x43 + x44 = c),where c∈ F*q.In this paper,by using the Jacobi sums and an analog of the Hasse-Davenport theorem,we obtain the exact formula for N(x41 +⋯+ x4n= c)and thus extend the known results.
Finite fieldRational pointDiagonal equationJacobi sum
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