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

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.

On the Fourth Hybrid Power Mean Involving the Generalized Gauss Sums

“…Tere are a number of studies with regard to the Gauss sums and their recursion properties, which can be found in [6][7][8][9][10][11][12][13][14][15].…”