BOUNDS ON FEWNOMIAL EXPONENTIAL SUMS OVER Z p TODD
نویسنده
چکیده
We obtain a number of new bounds for exponential sums of the type S(χ, f) = Pp−1 x=1 χ(x)ep(f(x)), with p a prime, f(x) = Pr i=1 aix ki , 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) = Pm i=1 aix ki + g(xd) with d|(p − 1) then |S(χ, f)| ≤ p (k1 · · · km) 1 m2 /d 1 2m . If f(x) = axk + h(xd) with d|(p− 1) and (k, p− 1) = 1 then |S(χ, f)| ≤ p/ √ d, and if f(x) = axk + bx−k + h(xd) with d|(p − 1) and (k, p− 1) = 1 then |S(χ, f)| ≤ p/ √ d+ √ 2p3/4.
منابع مشابه
Bounds on Fewnomial Exponential Sums Over
We obtain a number of new bounds for exponential sums of the type S(χ, f) = ∑p−1 x=1 χ(x)ep(f(x)), with p a prime, f(x) = ∑r i=1 aix ki , 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) = ∑m i=1 aix...
متن کاملA-Discriminants for Complex Exponents and Counting Real Isotopy Types
We take a first step toward extending the theory of A-discriminants, and Kapranov’s parametrization of A-discriminant varieties [Kap91], to a broader family of functions including polynomials as a very special case. As an application, we prove a quadratic upper bound on the number of isotopy types of real zero sets of certain n-variate exponential sums, in a setting where the best previous boun...
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 کاملBounds on Exponential Sums and the Polynomial Waring Problem Mod
Estimates are given for the exponential sum ∑p x=1 exp(2πif(x)/p), p a prime and f a nonzero integer polynomial, of interest in cases where the Weil bound is worse than trivial. The results extend those of Konyagin for monomials to a general polynomial. Such bounds readily yield estimates for the corresponding polynomial Waring problem mod p, namely the smallest γ such that f(x1)+ . . .+f(xγ) ≡...
متن کاملUpper Bounds on a Two-term Exponential Sum
We obtain upper bounds for mixed exponential sums of the type S(χ, f, pm) = ∑pm x=1 χ(x)epm (ax n+bx) where pm is a prime power with m ≥ 2 and χ is a multiplicative character (mod pm). If χ is primitive or p (a, b) then we obtain |S(χ, f, pm)| ≤ 2np 2 3 . If χ is of conductor p and p (a, b) then we get the stronger bound |S(χ, f, pm)| ≤ npm/2.
متن کامل