Estimates for Mixed Character Sums
نویسنده
چکیده
a nontrivial additive character of k. We are given a polynomial f = f(x1, ..., xn) in n ≥ 1 variables over k of degree d ≥ 1 which is a “Deligne polynomial”, i.e., its degree d is prime to p and its highest degree term, say fd, is a homogeneous form of degree d in n variables which is nonzero, and whose vanishing, if n ≥ 2, defines a smooth hypersurface in the projective space Pn−1. For a Deligne polynomial f as above, one has Deligne’s fundamental estimate [De-Weil I, 8.4]
منابع مشابه
Burgess bounds for short mixed character sums
This paper proves nontrivial bounds for short mixed character sums by introducing estimates for Vinogradov’s mean value theorem into a version of the Burgess method.
متن کاملEstimates for Nonsingular Mixed Character Sums
a nontrivial multiplicative character of k. We extend χ to k by defining χ(0) = 0. We wish to consider character sums over An, n ≥ 1, of the following form. We are given a polynomial f (x) := f (x1, . . . , xn) in k[x1, . . . ,Xn] of degree d ≥ 1, and we are given a second polynomial g(X) := g(x1, . . . , xn) in k[x1, . . . , xn] of degree e ≥ 1. We are interested in understanding when the sum ...
متن کاملA Survey on Pure and Mixed Exponential Sums modulo Prime Powers
where p is a prime power, epm(·) is the additive character epm(x) = e m and χ is a multiplicative character (mod p). The goals of this paper are threefold; first, to point out the similarity between exponential sums over finite fields and exponential sums over residue class rings (mod p) with m ≥ 2; second, to show how mixed exponential sums can be reduced to pure exponential sums when m ≥ 2 an...
متن کاملPure and Mixed Exponential Sums
We obtain explicit formulae and sharp estimates for pure and mixed exponential sums of the type S(f, pm) = ∑pm x=1 epm (f(x)) and S(χ, f, pm) = ∑pm x=1 χ(x)epm (f(x)), where p m is a prime power with m ≥ 2, χ is a multiplicative character (mod pm), and f is a polynomial. For nonconstant f (mod p) we prove |S(χ, f, pm)| ≤ 2d 1 M+1 p m(1− 1 M+1 ) where d is the degree of f and M is the maximum mu...
متن کاملCharacter sums with Beatty sequences on Burgess-type intervals
We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence {⌊αn + β⌋ : n = 1, 2, . . . }, where α, β ∈ R, and α is irrational. Our bounds are nontrivial over the same short intervals for which the classical character sum estimates of Burgess have been established. 2000 Mathematics Subject Classification: 11B50, 11L40, 11T24
متن کامل