A partial improvement of the Ax–Katz theorem

Given a system of polynomial equations over a finite field, estimating the p-divisibility of the number of solutions of the system is a classical problem which has been studied extensively since Chevalley–Warning. The degrees of the polynomials concerned play a crucial role in such estimates. Instead of considering all the variables, we focus on the variables with lower degree and the isolated variables and find a partial improvement of the Ax–Katz theorem. Our result also generalizes, improves and unifies those recently obtained for a single polynomial.