Exponential sums and rational points on complete intersections

Shparlinski, Igor E.; Skorobogatov, Alexei N.

doi:10.1112/s0025579300012912

Cited by 25 publications

(15 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Shparlinski and Skorobogatov [16] estimated the modulus of exponential sums over the variety of dimension n − s defined by a system of s forms in n variables, with a linear form in the exponent. They also applied the estimations to the study of the distribution of rational points on such a variety defined over a finite field or the field of rationals.…”

Section: Further Discussionmentioning

confidence: 99%

On a uniformly distributed phenomenon in matrix groups

Hu¹,

Li²

2013

Journal of Number Theory

View full text Add to dashboard Cite

show abstract

Section: Further Discussionmentioning

confidence: 99%

On a uniformly distributed phenomenon in matrix groups

Hu¹,

Li²

2013

Journal of Number Theory

View full text Add to dashboard Cite

show abstract

“…Our work has been mainly motivated by the exponential sum estimates on abstractly given homogeneous varieties due to authors in [19].…”

Section: 2mentioning

confidence: 99%

“…In Section 4, we derive a result for averaging problems over general homogeneous varieties, where we adapt the standard analysis technique in [3] together with the results on exponential sums in [19], see Lemma 4.2 below. In fact, this result generalizes our main results related to H k .…”

Section: 4mentioning

confidence: 99%

Averaging Operators Over Homogeneous Varieties Over Finite Fields

2015

View full text Add to dashboard Cite

Abstract. In this paper we study the mapping properties of the averaging operator over a variety given by a system of homogeneous equations over a finite field. We obtain optimal results on the averaging problems over two dimensional varieties whose elements are common solutions of diagonal homogeneous equations. The proof is based on a careful study of algebraic and geometric properties of such varieties. In particular, we show that they are not contained in any hyperplane and are complete intersections. We also address partial results on averaging problems over arbitrary dimensional homogeneous varieties which are smooth away from the origin.

show abstract

“…. , m, in m variables with integer coefficients, modulo a prime p, see [4,5,8,11,12]. In particular, subject some additional condition (related to the socalled A-number), Fouvry and Katz [5, Corollary 1.5] have given an asymptotic formula for the number of solutions to (1) in a box (x 1 , .…”

Section: Introductionmentioning

confidence: 99%

On Solutions to Some Polynomial Congruences in Small Boxes

Shparlinski¹

2013

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

We 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 solution 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.2010 Mathematics Subject Classification. 11D79, 11K38.

show abstract

Exponential sums and rational points on complete intersections

Cited by 25 publications

References 12 publications

On a uniformly distributed phenomenon in matrix groups

On a uniformly distributed phenomenon in matrix groups

Averaging Operators Over Homogeneous Varieties Over Finite Fields

On Solutions to Some Polynomial Congruences in Small Boxes

Contact Info

Product

Resources

About