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. 
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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
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Broberg , A note on a paper by R . Heath - Brown : ‘ ‘ The density of rational points on curves and surfaces
  • J . reine angew . Math .
  • 2004
A Note on the Distribution of Rational Points on Threefolds
Heath - Brown , The density of rational points on curves and surfaces
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