On the number of zeros to the equation f(x1)+... + f(xn)=a over finite fields

• Published in: Finite fields and their applications Vol. 76; p. 101922
• Main Authors: Zhu, Chaoxi, Feng, Yulu, Hong, Shaofang, Zhao, Junyong
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2021
• Subjects: Exponential sum; Generating series; Minimal polynomial; Rationality; Zero; Generating series; Zero; Rationality; Exponential sum; Minimal polynomial; 11T24; 11T23
• ISSN: 1071-5797; 1090-2465
• DOI: 10.1016/j.ffa.2021.101922

Let p be a prime, k a positive integer and let Fq be the finite field of q=pk elements. Let f(x) be a polynomial over Fq and a∈Fq. We denote by Ns(f,a) the number of zeros of f(x1)+⋯+f(xs)=a. In this paper, we show that∑s=1∞Ns(f,0)xs=x1−qx−xMf′(x)qMf(x), where Mf′(x) stands for the derivative of Mf(x) andMf(x):=∏m∈Fq⁎Sf,m≠0(x−1Sf,m) with Sf,m:=∑x∈FqζpTr(mf(x)), ζp being the p-th primitive unit root and Tr being the trace map from Fq to Fp. This extends Richman's theorem which treats the case of f(x) being a monomial. Moreover, we show that the generating series ∑s=1∞Ns(f,a)xs is a rational function in x and also present its explicit expression in terms of the first 2d+1 initial values N1(f,a),...,N2d+1(f,a), where d is a positive integer no more than q−1. From this result, the theorems of Chowla-Cowles-Cowles and of Myerson can be derived.