  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.
