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Sums and differences of four kth powers

Oscar Marmon (Institutionen för matematiska vetenskaper, matematik)
Monatshefte Fur Mathematik (0026-9255). Vol. 164 (2011), 1, p. 55-74.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: Sum of kth powers, Determinant method, Diagonal form, Integral points, counting rational-points, integral points, number, representations, density, hypersurfaces, varieties, cubes, barre o, 2001

Denna post skapades 2011-10-12.
CPL Pubid: 147158


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Institutionen för matematiska vetenskaper, matematik (2005-2016)



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