On the Character Sum of Polynomials and the Two-term Exponential Sums

2020 ◽  
Vol 36 (2) ◽  
pp. 196-206
Author(s):  
Xing Xing Lv ◽  
Wen Peng Zhang
Keyword(s):  
Exponential Sums ◽  
Character Sum

scholarly journals ON SOLUTIONS TO SOME POLYNOMIAL CONGRUENCES IN SMALL BOXES

2013 ◽  
Vol 89 (2) ◽  
pp. 300-307
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Exponential Sums ◽  
Polynomial Systems ◽  
Side Length ◽  
Character Sum ◽  
Number Of Solutions ◽  
Polynomial Congruences

AbstractWe use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

ON THE DISTRIBUTION OF POINTS ON THE GENERALIZED MARKOFF–HURWITZ AND DWORK HYPERSURFACES

2014 ◽  
Vol 10 (01) ◽  
pp. 151-160 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Exponential Sums ◽  
Side Length ◽  
Character Sum ◽  
Distribution Of Points

We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface [Formula: see text] for some polynomials fi ∈ ℤ[X] that are not constant modulo a prime p and integers ki with gcd (ki, p-1) = 1, i = 1, …, n. In the case of [Formula: see text] the above congruence is known as the Markoff–Hurwitz hypersurface, while for [Formula: see text] it is known as the Dwork hypersurface. In particular, we obtain non-trivial results about the number of solution in boxes with the side length below p1/2, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

scholarly journals The character sum of polynomials with k variables and two-term exponential sums

2021 ◽  
Vol 27 (1) ◽  
pp. 112-124
Author(s):  
Xiaoling Xu ◽  
◽  
Jiafan Zhang ◽  
Wenpeng Zhang ◽  
◽  
...  
Keyword(s):  
Asymptotic Formula ◽  
Exponential Sums ◽  
Character Sums ◽  
Gauss Sums ◽  
Character Sum ◽  
Power Mean ◽  
Computational Problem ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

The main purpose of this paper is using the properties of the classical Gauss sums and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sums of polynomials with k variables and the two-term exponential sums, and give an identity and asymptotic formula for it.

scholarly journals A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenpeng Zhang ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Gauss Sums ◽  
Power Mean ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

scholarly journals On Waring’s problem: Three cubes and a sixth power

2001 ◽  
Vol 163 ◽  
pp. 13-53 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley
Keyword(s):  
Exponential Sums ◽  
Large Integer ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Smooth Numbers ◽  
Waring's Problem ◽  
Order Of Magnitude ◽  
Almost All ◽  
Hardy Littlewood Method

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

2014 ◽  
Vol 13 (06) ◽  
pp. 1450013 ◽  
Author(s):  
Francis N. Castro ◽  
Ivelisse M. Rubio
Keyword(s):  
Exponential Sums ◽  
Polynomial Equations ◽  
Prime Field ◽  
Elementary Method ◽  
Covering Method ◽  
Systems Of Polynomial Equations

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

Short cubic exponential sums over primes

2017 ◽  
Vol 296 (1) ◽  
pp. 211-233
Author(s):  
Z. Kh. Rakhmonov ◽  
F. Z. Rakhmonov
Keyword(s):  
Exponential Sums ◽  
Exponential Sums Over Primes

scholarly journals An improved Mordell type bound for exponential sums

2004 ◽  
Vol 133 (2) ◽  
pp. 313-320 ◽  
Author(s):  
Todd Cochrane ◽  
Christopher Pinner
Keyword(s):  
Exponential Sums
Sign in / Sign up

Export Citation Format

Share Document