On Upper Bounds of Chalk and Hua for Exponential Sums

Let f be a polynomial of degree d with integer coefficients, p any prime, m any positive integer and S(f, pm) the exponential sum S(f, pm) = ∑pm x=1 epm(f(x)). We establish that if f is nonconstant when read (mod p), then |S(f, pm)| ≤ 4.41pm(1− 1 d . Let t = ordp(f ′), let α be a zero of the congruence p−tf ′(x) ≡ 0 (mod p) of multiplicity ν and let Sα(f, pm) be the sum S(f, pm) with x restricted to values congruent to α (mod pm). We obtain |Sα(f, pm)| ≤ min{ν, 3.06}p t ν+1 p m(1− 1 ν+1 ) for p odd, m ≥ t+2 and dp(f) ≥ 1. If, in addition, p ≥ (d − 1)(2d)/(d−2), then we obtain the sharp upper bound |Sα(f, pm)| ≤ p 1 ν+1 .