On the number of solutions to certain diagonal equations over finite fieldsстатья
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Аннотация:We consider a diagonal equation, which can be reduced to the form
x_1^2 +⋯+x_{n-2}^2 +x_{n-1}^ k +x_n^k =0
over a finite field of characteristic p>2. In 1997, Sun obtained the explicit formula for the number of solutions to an equation of this type when n is even. In this paper, we find explicit formulas for the number of solutions when n is odd, k=(2^r)h, and there exists a positive integer ℓ such that
p^ℓ ≡2^{m-1} h+1(mod (2^m) h), m=3 or 4, r≥m, h=1 or 3.