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Monatshefte für Mathematik

, Volume 164, Issue 1, pp 55–74 | Cite as

Sums and differences of four kth powers

  • Oscar Marmon
Article
  • 63 Downloads

Abstract

We prove an upper bound for the number of representations of a positive integer N as the sum of four kth powers of integers of size at most B, using a new version of the determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form \({O_{N}(B^{c/\sqrt{k}})}\), whereas earlier versions of the determinant method would produce an exponent for B of order k −1/3 (uniformly in N) in this case. Furthermore, we prove that the number of representations of a positive integer N as a sum of four kth powers of non-negative integers is at most \({O_{\varepsilon}(N^{1/k+2/k^{3/2}+\varepsilon})}\) for k ≥ 3, improving upon bounds by Wisdom.

Keywords

Sum of kth powers Determinant method Diagonal form Integral points 

Mathematics Subject Classification (2000)

11D85 14G05 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematical SciencesChalmers University of TechnologyGothenburgSweden
  2. 2.Mathematical SciencesUniversity of GothenburgGothenburgSweden

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