# Counting rational points on hypersurfaces

@inproceedings{Browning2004CountingRP, title={Counting rational points on hypersurfaces}, author={Tim D. Browning and D. R. Heath-Brown}, year={2004} }

Abstract For any n ≧ 2, let F ∈ ℤ [ x 1, … , xn ] be a form of degree d≧ 2, which produces a geometrically irreducible hypersurface in ℙn–1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F; B) = O(B n− 2+ ε ), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d, n and ε.

#### 48 Citations

On the density of rational and integral points on algebraic varieties

- Mathematics
- 2005

Abstract Let X ⊂ ℙ n be a projective geometrically integral variety over of dimension r and degree d ≧ 4. Suppose that there are only finitely many (r − 1)-planes over on X. The main result of this… Expand

Counting rational points on cubic hypersurfaces

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Let X be a geometrically integral projective cubic hypersurface defined over the rationals, with dimension D and singular locus of dimension at most D-4. For any \epsilon>0, we show that X contains… Expand

O ct 2 00 4 Counting Rational Points on Algebraic Varieties

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For any N ≥ 2, let Z ⊂ P be a geometrically integral algebraic variety of degree d. This paper is concerned with the number NZ(B) of Q-rational points on V which have height at most B. For any ε > 0… Expand

2 2 A pr 2 00 5 Counting Rational Points on Algebraic Varieties

- 2008

For any N ≥ 2, let Z ⊂ P be a geometrically integral algebraic variety of degree d. This paper is concerned with the number NZ(B) of Q-rational points on Z which have height at most B. For any ε > 0… Expand

The density of rational points near hypersurfaces

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We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies… Expand

Counting Multiplicities in a Hypersurface over a Number Field

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We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An… Expand

On uniform bounds for rational points on rational curves of arbitrary degree

- Mathematics
- 2013

Abstract We show that for any ϵ > 0 the number of rational points of height less than B on the image of a degree d map from P 1 to P 2 is bounded above by C d B 2 / d + d 2 , where the point is that… Expand

Density of rational points on a quadric bundle in P 3 × P 3

- Mathematics
- 2018

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface… Expand

The Density of Rational Points on Non-Singular Hypersurfaces, I

- Mathematics
- 2005

LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xeℤ n ;F(x)=0, |x|⩽X}, where\(\left| x \right| = \mathop {max}\limits_{1 \leqslant r \leqslant n} \left| {x_r }… Expand

COUNTING RATIONAL POINTS ON CUBIC

- 2008

Let X ⊂ P be a geometrically integral cubic hypersurface defined over Q, with singular locus of dimension 6 dim X − 4. Then the main result in this paper is a proof of the fact that X(Q) contains… Expand

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