Sums and differences of four kth powers
Journal article, 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
Author
Oscar Marmon
Chalmers, Mathematical Sciences, Mathematics
University of Gothenburg
Monatshefte für Mathematik
0026-9255 (ISSN) 1436-5081 (eISSN)
Vol. 164 1 55-74Subject Categories
Mathematics
DOI
10.1007/s00605-010-0248-2