scholarly journals p-adic Valuation of Exponential Sums in One Variable Associated to Binomials

2017 ◽  
Vol 12 (1) ◽  
pp. 37-53 ◽  
Author(s):  
Francis N. Castro ◽  
Raúl Figueroa ◽  
Puhua Guan
Keyword(s):  
Lower Bound ◽  
Exponential Sums ◽  
Adic Valuation ◽  
Value Sets

Abstract In this paper we compute the p-adic valuation of exponential sums associated to binomials F(X) = aXd₁ + bXd₂ over Fp. In particular, its p-adic valuation is constant for a, b ∈ F∗p . As a byproduct of our results, we obtain a lower bound for the sizes of value sets of binomials over Fq.

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

A lower bound for the L 1 norm of exponential sums

Mathematika ◽  
1974 ◽  
Vol 21 (2) ◽  
pp. 155-159 ◽  
Author(s):  
S. K. Pichorides
Keyword(s):  
Lower Bound ◽  
Exponential Sums

On the weak forms of the 2-part of Birch and Swinnerton-Dyer conjecture

2018 ◽  
Vol 168 (1) ◽  
pp. 197-209 ◽  
Author(s):  
SHUAI ZHAI
Keyword(s):  
Lower Bound ◽  
Elliptic Curves ◽  
Quadratic Twists ◽  
Central Value ◽  
The Family ◽  
Adic Valuation

AbstractIn this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$.

On thep-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of

2016 ◽  
Vol 164 (1) ◽  
pp. 67-98 ◽  
Author(s):  
YUKAKO KEZUKA
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Lower Bounds ◽  
Complex Multiplication ◽  
Ring Of Integers ◽  
Quadratic Twists ◽  
The Family ◽  
Adic Valuation

AbstractWe study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

scholarly journals The Maximal Difference of Different Powers of an Element Modulo n

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Jinyun Qi ◽  
Zhefeng Xu
Keyword(s):  
Lower Bound ◽  
Exponential Sums

In this paper, we investigate the maximal difference of integer powers of an element modulo n . Let a n denote the integer b with 1 ≤ b ≤ n such that a ≡ b mod   n for any integer a . Using the bounds for exponential sums, we obtain a lower bound of the function H m 1 , m 2 n : = max a m 1 n − a m 2 n : 1 ≤ a ≤ n , a , n = 1 , which gives n − H m 1 , m 2 n = O n 3 / 4 + o 1 .

The L1-norm of exponential sums in d

2013 ◽  
Vol 154 (3) ◽  
pp. 381-392 ◽  
Author(s):  
GIORGIS PETRIDIS
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Absolute Constant ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Correct Order ◽  
L1 Norm ◽  
Finite Set ◽  
Order Of Magnitude ◽  
Technical Sense

AbstractLet A be a finite set of integers and FA(x) = ∑a∈A exp(2πiax) be its exponential sum. McGehee, Pigno and Smith and Konyagin have independently proved that ∥FA∥1 ≥ c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L1-norm of exponential sums of sets in the d-dimensional grid d. We show that ∥FA∥1 is considerably larger than log|A| when A ⊂ d has multidimensional structure. We furthermore prove similar lower bounds for sets in , which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno and Smith and Konyagin.

Lower bound of overall exponential sums for polynomialsϕ(x d )

1989 ◽  
Vol 20 (4) ◽  
pp. 279-287
Author(s):  
L. A. Bassalygo ◽  
V. A. Zinov'ev ◽  
S. N. Licyn
Keyword(s):  
Lower Bound ◽  
Exponential Sums
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