Sums and differences of four kth powers
Artikel i vetenskaplig tidskrift, 2011

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/root 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(epsilon)(N(1/k+2/k3/2+epsilon)) for k >= 3, improving upon bounds by Wisdom.

Sum of kth powers

integral points

hypersurfaces

representations

varieties

2001

density

number

counting rational-points

barre o

Integral points

Determinant method

Diagonal form

cubes

Författare

Oscar Marmon

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

Monatshefte für Mathematik

0026-9255 (ISSN) 1436-5081 (eISSN)

Vol. 164 1 55-74

Ämneskategorier

Matematik

DOI

10.1007/s00605-010-0248-2