Optimal Divisibility for Certain Diagonal Equations Over Finite Fields

We improve the divisibility of diagonal equations of the type a1Xd1 1 + · · · + anXdn n = β over finite fields. This improvement gives cases where Ax’s, Moreno–Moreno’s and Wan’s results can be greatly improved. We prove that some of our estimates are the best possible. In particular, we compute the optimal divisibility for diagonal equations whenever di divides pj + 1. We present an algorithm in which we only need to do one simple computation to get the best divisibility of a family of diagonal equations with β = 0 for Fqm and we apply this in order to give the best p-adic Serre bound. Finally, we prove that every element of a finite field Fpf is the sum of two dth powers, when d divides pj + 1.