Estimation of exponential sums of polynomials of higher degrees II

Abstract

In this article we will prove certain identities between the above exponential sum and hyper-Kloosterman sums, generalize the above estimation for the exponential sum to other cases of m when a ≥ 2, and establish new bounds for hyper-Kloosterman sums. Write p ‖n if p |n but ph+1 -n. Theorem 1. Let p be a prime, q = p, a ≥ 2, and k an integer with a ≤ k < φ(q) and p k. We set h by p ‖ (k − 1). Then for any b and c relatively prime to p we have