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 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.

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 Local stationarity in exponential last-passage percolation

2021 ◽  
Author(s):  
Márton Balázs ◽  
Ofer Busani ◽  
Timo Seppäläinen
Keyword(s):  
Brownian Motion ◽  
Total Variation ◽  
High Probability ◽  
Probabilistic Methods ◽  
Side Length ◽  
Last Passage Percolation ◽  
Last Passage Times ◽  
Starting Point ◽  
Point To Point ◽  
Passage Times

AbstractWe consider point-to-point last-passage times to every vertex in a neighbourhood of size $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 at distance N from the starting point. The increments of the last-passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends only on $$\delta $$ δ . Through this result we show that (1) the $$\text {Airy}_2$$ Airy 2 process is locally close to a Brownian motion in total variation; (2) the tree of point-to-point geodesics from every vertex in a box of side length $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 going to a point at distance N agrees inside the box with the tree of semi-infinite geodesics going in the same direction; (3) two point-to-point geodesics started at distance $$N^{\nicefrac {2}{3}}$$ N 2 3 from each other, to a point at distance N, will not coalesce close to either endpoint on the scale N. Our main results rely on probabilistic methods only.

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