# The density of rational points on non-singular hypersurfaces, I

Browning TD, Heath Brown R. 2006. The density of rational points on non-singular hypersurfaces, I. Bulletin of the London Mathematical Society. 38(3), 401–410.

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Browning, Timothy D

^{IST Austria}^{}; Heath-Brown, RogerAbstract

For any n≥3, let F ∈ Z[X0,...,Xn ] be a form of degree d *≥5 that defines a non-singular hypersurface X ⊂ Pn . The main result in this paper is a proof of the fact that the number N (F ; B) of Q-rational points on X which have height at most B satisfiesN (F ; B) = Od,ε,n (Bn −1+ε ), for any ε > 0. The implied constant in this estimate depends at most upon d, ε and n. New estimates are also obtained for the number of representations of a positive integer as the sum of three dth powers, and for the paucity of integer solutions to equal sums of like polynomials.*

Publishing Year

Date Published

2006-12-23

Journal Title

Bulletin of the London Mathematical Society

Volume

38

Issue

3

Page

401 - 410

IST-REx-ID

### Cite this

Browning TD, Heath Brown R. The density of rational points on non-singular hypersurfaces, I.

*Bulletin of the London Mathematical Society*. 2006;38(3):401-410. doi:10.1112/S0024609305018412Browning, T. D., & Heath Brown, R. (2006). The density of rational points on non-singular hypersurfaces, I.

*Bulletin of the London Mathematical Society*. Wiley-Blackwell. https://doi.org/10.1112/S0024609305018412Browning, Timothy D, and Roger Heath Brown. “The Density of Rational Points on Non-Singular Hypersurfaces, I.”

*Bulletin of the London Mathematical Society*. Wiley-Blackwell, 2006. https://doi.org/10.1112/S0024609305018412.T. D. Browning and R. Heath Brown, “The density of rational points on non-singular hypersurfaces, I,”

*Bulletin of the London Mathematical Society*, vol. 38, no. 3. Wiley-Blackwell, pp. 401–410, 2006.Browning, Timothy D., and Roger Heath Brown. “The Density of Rational Points on Non-Singular Hypersurfaces, I.”

*Bulletin of the London Mathematical Society*, vol. 38, no. 3, Wiley-Blackwell, 2006, pp. 401–10, doi:10.1112/S0024609305018412.