Improvement to Moreno–Morenoʼs theorems

In this work, we introduce the p-weight degree of a polynomial over a finite field with respect to a subset of the variables. Using this p-weight, we improve the results of Moreno and Moreno for polynomial equations and for exponential sums over finite fields. We prove that our results cannot be improved in general because a family of polynomials where our bounds are attained is provided. Combining our result with a result of Cao and Sun, we give a p-adic improvement to the p-divisibility of general diagonal equations.