# Counting rational points on hypersurfaces

@inproceedings{Browning2004CountingRP,
title={Counting rational points on hypersurfaces},
author={Tim D. Browning and D. R. Heath-Brown},
year={2004}
}
• Published 2004
• Mathematics
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.
On the density of rational and integral points on algebraic varieties
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 thisExpand
Counting rational points on cubic hypersurfaces
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 containsExpand
O ct 2 00 4 Counting Rational Points on Algebraic Varieties
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 > 0Expand
2 2 A pr 2 00 5 Counting Rational Points on Algebraic Varieties
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 > 0Expand
The density of rational points near hypersurfaces
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 reliesExpand
Counting Multiplicities in a Hypersurface over a Number Field
• Mathematics
• 2017
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. AnExpand
On uniform bounds for rational points on rational curves of arbitrary degree
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 thatExpand
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 hypersurfaceExpand
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
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) containsExpand

#### References

SHOWING 1-10 OF 20 REFERENCES
Counting Rational Points On Threefolds
• Physics
• 2004
Let X ⊂ ℙ4 be an irreducible hypersurface and e > 0 be given. We show that there are O(B3+e), resp. O(B55/18+e), rational points on ℙ4 lying on X when X is of degree d ⩾ 4, resp. d = 3. The impliedExpand
The density of rational points on cubic surfaces
NF (P ) = N(P ) = #{x ∈ Z4 : F (x) = 0, |x| ≤ P}, where |x| is the Euclidean length of x. This paper is concerned with the behaviour of N(P ) as P tends to infinity. It is clear that if the surface FExpand
The density of rational points on curves and surfaces
• Mathematics
• 2002
Let $C$ be an irreducible projective curve of degree $d$ in $\mathbb{P}^3$, defined over $\overline{\mathbb{Q}}$. It is shown that $C$ has $O_{\varepsilon,d}(B^{2/d+\varepsilon})$ rational points ofExpand
Density of integral and rational points on varieties
© Société mathématique de France, 1995, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec lesExpand
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
Diophantine Approximations and Diophantine Equations
Siegel's lemma and heights.- Diophantine approximation.- The thue equation.- S-unit equations and hyperelliptic equations.- Diophantine equations in more than two variables.
Broberg , A note on a paper by R . Heath - Brown : ‘ ‘ The density of rational points on curves and surfaces
• J . reine angew . Math .
• 2004
Heath - Brown , The density of rational points on curves and surfaces
• Acta Arithmetica
• 1997