Bounds on Fewnomial Exponential Sums Over
نویسنده
چکیده
We obtain a number of new bounds for exponential sums of the type S(χ, f) = ∑p−1 x=1 χ(x)ep(f(x)), with p a prime, f(x) = ∑r i=1 aix ki , 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) = ∑m i=1 aix ki + g(xd) with d|(p − 1) then |S(χ, f)| ≤ p (k1 · · · km) 1 m2 /d 1 2m . If f(x) = axk + h(xd) with d|(p− 1) and (k, p− 1) = 1 then |S(χ, f)| ≤ p/ √ d, and if f(x) = axk + bx−k + h(xd) with d|(p − 1) and (k, p− 1) = 1 then |S(χ, f)| ≤ p/ √ d+ √ 2p3/4.
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