Sums of the higher divisor function of diagonal homogeneous forms in short intervals

Let $d_k(n)$ denote the k-th divisor function. In this paper, we give the asymptotic formula of the sum

$$\sum _{\begin{array}{c}x-y\le n_i^r\le x+y\\ i=1,2,\dots ,l\end{array}}{d}_k(n_1^r+n_2^r+\dots +n_l^r),$$

where $n_1,n_2,\dots ,n_l\in {\mathbb{Z}}^{+}$, $k\ge 2$, $r\ge 3$ and $l>2^{r-1}$ are integers.