Bounds on fewnomial exponential sums over ℤp

Abstract

AbstractWe obtain a number of new bounds for exponential sums of the type S(χ, f) = ∑x = 1p−1 χ(x) ep(f(x)), with p a prime, f(x) = ∑i = 1raixki, ai, ki ∈ ℤ, 1 ≤ i ≤ r and χ a multiplicative character (mod p). The bounds refine earlier Mordell-type estimates and are particularly effective for polynomials in which a certain number of the ki have a large gcd with p − 1. For instance, if f(x) = ∑i = 1maixki + g(xd) with d|(p − 1) then ${|S(\chi,f)| \le p\ (k_1\cdots k_m)^{\frac1{m^2}}/d^{\frac 1{2m}}$. If f(x) = axk + h(xd) with d|(p − 1) and (k, p − 1) = 1 then $|S(\chi,f)|\le p/\sqrt{d}$, and if f(x) = axk + bx−k + h(xd) with d|(p − 1) and (k, p − 1) = 1 then $|S(\chi,f)| \le p/\sqrt{d} + \sqrt{2} p^{3/4}$.