Improvements upon the Chevalley–Warning–Ax–Katz-type estimates

Let f be a polynomial over finite field Fq with q elements and let N(f=0) denote the number of zeros of f in Fq. The q-divisibility properties of N(f=0) have been studied by many authors, such as Chavelley, Warning, Ax, Katz, etc. In this paper, by reducing the degree of a given polynomial meanwhile remaining the number of its zeros unchanged, we present an improvement upon the Chevalley–Warning–Ax–Katz-type estimates in many cases. Furthermore, our result can improve Cao–Sun's reduction recently obtained on counting the number of zeros of general diagonal equations over finite fields.