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Bounds on fewnomial exponential sums over ℤp

Published online by Cambridge University Press:  22 June 2010

TODD COCHRANE  and
CHRISTOPHER PINNER
TODD COCHRANE
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A. e-mail: cochrane@math.ksu.edu, pinner@math.ksu.edu
CHRISTOPHER PINNER
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A. e-mail: cochrane@math.ksu.edu, pinner@math.ksu.edu
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Abstract

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 ≤ 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 . 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 .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 217 - 227
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

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