Rational points on complete intersections of higher degree, and mean values of Weyl sums

P Salberger, TD Wooley

Research output: Contribution to journalArticle (Academic Journal)peer-review

5 Citations (Scopus)

Abstract

It is known from Vaughan and Wooley's work on Waring's problem that every sufficiently large natural number is the sum of at most 17 fifth powers [13]. It is also known that at least six fifth powers are required to be able to express every sufficiently large natural number as a sum of fifth powers (see, for instance, [5, Theorem 394]). The techniques of [13] allow one to show that almost all natural numbers are the sum of nine fifth powers. A problem of related interest is to obtain an upper bound for the number of representations of a number as a sum of a fixed number of powers. Let R(n) denote the number of representations of the natural number n as a sum of four fifth powers. In this paper, we establish a non-trivial upper bound for R(n), which is expressed in the following theorem.
Translated title of the contributionRational points on complete intersections of higher degree, and mean values of Weyl sums
Original languageEnglish
Pages (from-to)317 - 342
Number of pages26
JournalJournal of the London Mathematical Society
Volume82, number 2
DOIs
Publication statusPublished - Jul 2010

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