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Bounds on fewnomial exponential sums over ℤp

Published online by Cambridge University Press:  22 June 2010

Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A. e-mail:,
Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A. e-mail:,


We 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 ≤ ir 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 . If f(x) = axk + h(xd) with d|(p − 1) and (k, p − 1) = 1 then , and if f(x) = axk + bx−k + h(xd) with d|(p − 1) and (k, p − 1) = 1 then .

Research Article
Copyright © Cambridge Philosophical Society 2010

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