Using Stepanov’s Method for Exponential Sums Involving Rational Functions
نویسندگان
چکیده
For a non-trivial additive character ψ and multiplicative character χ on a finite field Fq , and rational functions f, g in Fq(x), we show that the elementary Stepanov-Schmidt method can be used to obtain the corresponding Weil bound for the sum ∑ x∈Fq\S χ(g(x))ψ(f(x)) where S is the set of the poles of f and g. We also determine precisely the number of characteristic values ωi of modulus q1/2 and the number of modulus 1.
منابع مشابه
EXPONENTIAL SUM ESTIMATES OVER SUBGROUPS AND ALMOST SUBGROUPS OF Zq, WHERE q IS COMPOSITE WITH FEW PRIME FACTORS
In this paper we extend the exponential sum results from [B-K] and [B-G-K] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size q, for any given δ > 0. The method consists in first establishing a ‘sum-product theorem’ for general subsets A of Z. If q...
متن کاملStepanov’s Method Applied to Binomial Exponential Sums
For a prime p and binomial axk+bxl with 1 ≤ l < k < 1 32 (p−1) 2 3 , we use Stepanov’s method to obtain the bound ∣∣∣∣∣ p−1 ∑ x=1 ep(ax k + bx) ∣∣∣∣∣ max { 1, l∆− 1 3 } 1 4 k 1 4 p 3 4 , where ∆ = k−l (k,l,p−1) .
متن کاملNumerical solution of Troesch's problem using Christov rational functions
We present a collocation method to obtain the approximate solution of Troesch's problem which arises in the confinement of a plasma column by radiation pressure and applied physics. By using the Christov rational functions and collocation points, this method transforms Troesch's problem into a system of nonlinear algebraic equations. The rate of convergence is shown to be exponential. The numer...
متن کاملA Hybrid Mean Value Involving Dedekind Sums and the General Exponential Sums
The main purpose of this paper is using the analytic method, A. Weil's classical work for the upper bound estimate of the general exponential sums, and the properties of Gauss sums to study the hybrid mean value problem involving Dedekind sums and the general exponential sums and give a sharp asymptotic formula for it.
متن کاملL-functions of Exponential Sums over One-dimensional Affinoids: Newton over Hodge
This paper proves a sharp lower bound for Newton polygons of L-functions of exponential sums of one-variable rational functions. Let p be a prime and let Fp be the algebraic closure of the finite field of p elements. Let f(x) be any one-variable rational function over Fp with l poles of orders d1, . . . , dl. Suppose p is coprime to d1 · · · dl. We prove that there exists a tight lower bound wh...
متن کامل