On the number of zeros of diagonal quartic forms over finite fields

Abstract

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 {x_{1}^{4}+\cdots+x_{n-1}^{4}+yx_{n}^{4}=0} , respectively. In 1979, Myerson used Gauss sums and exponential sums to show that the generating function ∑ n = 1 ∞ N n ⁢ ( 0 ) ⁢ x n {\sum_{n=1}^{\infty}N_{n}(0)x^{n}} is a rational function in x and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions ∑ n = 1 ∞ N n ⁢ ( c ) ⁢ x n and ∑ n = 1 ∞ M n + 1 ⁢ ( y ) ⁢ x n \sum_{n=1}^{\infty}N_{n}(c)x^{n}\quad\text{and}\quad\sum_{n=1}^{\infty}M_{n+1}% (y)x^{n} are rational functions in x. We also obtain the explicit expressions of these generating functions. Our result extends Myerson’s theorem gotten in 1979.