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Rational points on complete intersections of higher degree, and mean values of Weyl sums
Author(s) -
Salberger Per,
Wooley Trevor D.
Publication year - 2010
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq027
Subject(s) - degree (music) , mathematics , diagonal , dimension (graph theory) , bounded function , intersection (aeronautics) , rational function , function (biology) , pure mathematics , combinatorics , complete intersection , square root , mathematical analysis , geometry , physics , evolutionary biology , acoustics , engineering , biology , aerospace engineering
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 equationx 1 d + ⋯ + x s d = x s + 1 d + ⋯ + x 2 s d , provided that d ⩾ (2 s ) 4 s . 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.