The number of solutions of certain diagonal equations over finite fields

On the solutions of the equation 𝑥^{𝑚}+𝑦^{𝑚}-𝑧^{𝑚}=1 in a finite field

“…In a more general setting Hua-Vandiver [7] as well as Weil [18] give formulas for the number of solutions involving Jacobi sums (see also [9, Chapter 8, Theorem 5] for a comprehensive exposition and [10] for more literature). However, these formulas are hard to evaluate for large m. Explicit and simple formulas are only known for certain special cases, namely when m is small [5,13], when k is small [17,13], or when 2\s and m\(^/q + 1) ( [14,8,6] and, more general, [19]). …”

Rational points and zeta functions of some curves over finite fields

“…So far as we know, there are handful classes of curves for which N q could be calculated explicitly. See [6,15,16] for details.…”

“…has been investigated [16]. Our main tool is the explicit evaluating of some exponential sums which will be introduced below.…”

Some Systems of Diagonal Equations over Finite Fields

“…Equations of (S) are homogeneous ones and therefore, this system defines a projective variety over % O whose number NR of rational projective points over % O I is NR"(N!1)(qI!1)\. In the case of only one equation (r"1), results are given in [4,5] In Section 2 we recall classical facts of the theory of linear codes. The reference is [3].…”