Abstract
We establish upper bounds for the number of rational points of bounded height on complete intersections. When the degree of the intersection is sufficiently large in terms of its dimension, and the contribution arising from appropriate linear spaces is removed, these bounds are smaller than those arising from the expectation of ‘square-root cancellation’. In particular, there is a paucity of non-diagonal solutions to the equation x 1 d + ⋯ + x s d = x s + 1 d + ⋯ + x 2 s d , provided that d ⩾ (2s)4s. There are consequences for the approximate distribution function of Weyl sums of higher degree, and also for quasi-diagonal behaviour in mean values of smooth Weyl sums.
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