On the number of zeros of diagonal cubic forms over finite fields
Shaofang Hong, Chaoxi Zhu · Forum Mathematicum · 2021
Abstract Let 𝔽 q {\mathbb{F}_{q}} be the finite field of q = p k {q=p^{k}} elements with p being a prime and let k be a positive integer. For any y , z ∈ 𝔽 q {y,z\in\mathbb{F}_{q}} , let N s ( z ) {N_{s}(z)} and T s ( y ) {T_{s}(y)} denote the numbers of zeros of x 1 3 + ⋯ + x s 3 = z {x_{1}^{3}+\cdots+x_{s}^{3}=z} and x 1 3 + ⋯ + x s - 1 3 + y x s 3 = 0 {x_{1}^{3}+\cdots+x_{s-1}^{3}+yx_{s}^{3}=0} , respectively. Gauss proved that if q = p {q=p} , p ≡