Bounds on complete exponential sums
Todd Cochrane · Birkhäuser Boston eBooks · 1996
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 ℙ n-l, then the characteristic values of S(g)have weight ≤ n + ℓ + 1, and so $$\left| {S(g)} \right|(4d + 5)^n p^{\frac{{n + l + 1}}{2}}$$ .