Counting Rational Points on Hypersurfaces

For any n ≥ 2, let F ∈ Z[x1, . . . , xn] be a form of degree d ≥ 2, which produces a geometrically irreducible hypersurface in P. 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), 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.