Counting rational points on hypersurfaces and higher order expansions

Schindler, Damaris

doi:10.1016/j.jnt.2016.09.001

Cited by 2 publications

(2 citation statements)

References 14 publications

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“…Theorem in the special case

F = Q

and

n = 4

recovers her result, with a better error term, but with less explicit analysis of the contribution of ramified places (including

\infty

). Vaughn and Wooley and Schindler have investigated secondary and higher order terms in Waring's problem. However, these terms appear for an entirely different reason, namely that in these works they estimate the number of zeros in a suitable box instead of a smoothed box using a Schwartz function

f

as above.…”

Section: Introductionmentioning

confidence: 99%

Secondary terms in asymptotics for the number of zeros of quadratic forms over number fields

Getz

2018

J. London Math. Soc.

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Let Q be a nondegenerate quadratic form on a vector space V of even dimension n over a number field F. Via the circle method or automorphic methods one can give good estimates for smoothed sums over the number of zeros of the quadratic form whose coordinates are of size at most X (properly interpreted). For example, when F=Q and dimV>4 Heath‐Brown has given an asymptotic of the form: c1Xn−2+OQ,ε,ffalse(Xn/2+εfalse), for any ε>0. Here c1∈C and f∈S(V(R)) is a smoothing function. We refine Heath‐Brown's work to give an asymptotic of the form: c1Xn−2+c2Xn/2+OQ,ε,ffalse(Xn/2+ε−1false) over any number field. Here c2∈C. Interestingly the secondary term c2 is the sum of a rapidly decreasing function on V(R) over the zeros of Q∨, the form whose matrix is inverse to the matrix of Q. We also prove analogous results in the boundary case n=4, generalizing and refining Heath‐Brown's work in the case F=Q.

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“…Theorem in the special case

F = Q

and

n = 4

recovers her result, with a better error term, but with less explicit analysis of the contribution of ramified places (including

\infty

f

as above.…”

Section: Introductionmentioning

confidence: 99%

Secondary terms in asymptotics for the number of zeros of quadratic forms over number fields

Getz

2018

J. London Math. Soc.

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“…In most cases, one only obtains the main term in the asymptotic. This changed in the recent papers of Schindler, Getz, Vaughn and Wooley (see [Sch17,Get18,VW18]). In this paper, we continue this investigation on quadratic form following the Health-Brown's new version of the circle method, which is Department of Mathematics, Duke University, tran2015@math.duke.edu.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Secondary terms in asymptotics for the number of zeros of quadratic forms

Huong¹

2019

Preprint

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Let F be a non-degenerate quadratic form on an n-dimensional vector space V over the rational numbers. One is interested in counting the number of zeros of the quadratic form whose coordinates are restricted in a smoothed box of size B, roughly speaking. For example, Heath-Brown gave an asymptotic of the form: c 1 B n−2 +O J,ǫ,ω (B (n−1)/2+ǫ ), for any ǫ > 0 and dimV ≥ 5, where c 1 ∈ C and ω ∈ S(V (R)) is a smooth function. More recently, Getz gave an asymptotic of the form: c 1 B n−2 + c 2 B n/2 + O J,ǫ,ω (B n/2+ǫ−1 ) when n is even, in which c 2 ∈ C has a pleasant geometric interpretation. We consider the case where n is odd and give an analogous asymptotic of the form: c 1 B n−2 +c 2 B (n−1)/2 +O J,ǫ,ω (B n/2+ǫ−1 ). Notably it turns out that the geometric interpretation of the constant c 2 of the asymptotic in the odd degree and even degree cases is strikingly different.Remark 1. Before we dive into our investigation, we want to make some comments here.

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Counting rational points on hypersurfaces and higher order expansions

Cited by 2 publications

References 14 publications

Secondary terms in asymptotics for the number of zeros of quadratic forms over number fields

Secondary terms in asymptotics for the number of zeros of quadratic forms over number fields

Secondary terms in asymptotics for the number of zeros of quadratic forms

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