On upper bounds of Chalk and Hua for exponential sums

Abstract

Abstract. Let f be a polynomial of degree d with integer coefficients, p any prime, m any positive integer and) . Let t = ordp(f ), let α be a zero of the congruence p −t f (x) ≡ 0 (mod p) of multiplicity ν and let Sα(f, p m ) be the sum S(f, p m ) with x restricted to values congruent to α (mod p m ). We obtain) for p odd, m ≥ t+2 and dp(f ) ≥ 1. ) .