Abstract
We give a brief survey on complete exponential sums of the type S(g) = Σx e p (g(x)), with g a polynomial in n variables over the finite field 픽 p , and prove several new results including the following. If g is non-composite, that is, not of the type g(x) = f(h(x)) with deg f ≥ 2, and p ≥ c 1(d, n)then the weights of the characteristic values of S(g) are all ≤ 2n - 2. For homogeneous g we can take c 1(d,n) = 2. A partial converse is also given. Next, if g is homogeneous, absolutely irreducible of degree d, and has a singular locus of dimension ℓ in ℙn-l, then the characteristic values of S(g)have weight ≤ n + ℓ + 1, and so \(\left| {S(g)} \right|(4d + 5)^n p^{\frac{{n + l + 1}}{2}}\).
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© 1996 Birkhäuser Boston
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Cochrane, T. (1996). Bounds on complete exponential sums. In: Berndt, B.C., Diamond, H.G., Hildebrand, A.J. (eds) Analytic Number Theory. Progress in Mathematics, vol 138. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4086-0_11
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DOI: https://doi.org/10.1007/978-1-4612-4086-0_11
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