On the number of zeros of diagonal quartic forms over finite fields
Junyong Zhao, Yulu Feng, Shaofang Hong, Chaoxi Zhu · Forum Mathematicum · 2022
Abstract Let 𝔽 q {\mathbb{F}_{q}} be the finite field of q = p m ≡ 1 ( mod 4 ) {q=p^{m}\equiv 1~{}(\bmod~{}4)} elements with p being an odd prime and m being a positive integer. For c , y ∈ 𝔽 q {c,y\in\mathbb{F}_{q}} with y ∈ 𝔽 q * {y\in\mathbb{F}_{q}^{*}} non-quartic, let N n ( c ) {N_{n}(c)} and M n ( y ) {M_{n}(y)} be the numbers of zeros of x 1 4 + ⋯ + x n 4 = c {x_{1}^{4}+\cdots+x_{n}^{4}=c} and x 1 4 + ⋯ + x n - 1 4 + y x n 4 = 0