Sums and differences of four k-th powers

Details Sums and differences of four k-th powers Sums and differences of four k-th powers

2009-12-08

arxiv.org/abs/0912.1527v2

Mathematics

Number Theory

18 pages. Mistake corrected in the statement of Theorem 1.2. To appear in Monatsh. Math

Scientific paper

10.1007/s00605-010-0248-2

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th 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 $k^{-1/3}$ in this case. Furthermore, we prove that the number of representations of a positive integer $N$ as a sum of four $k$-th powers of non-negative integers is at most $O_{\epsilon}(N^{1/k+2/k^{3/2}+\epsilon})$ for $k \geq 3$, improving upon bounds by Wisdom.

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