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

By Tim D Browning, D. R. Heath-Brown. 2005

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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 satisfies N(F; B)=Od,ɛ,n(Bn−1+ɛ) , for any ɛ > 0. The implied constant in this estimate depends at most upon d, ɛ 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. 2000 Mathematics Subject Classification 11G35 (primary), 11P05, 14G05 (secondary).

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

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