Decoupling for finite type phases in higher dimensions

In this paper, we establish an $\ell^2$ decoupling inequality for the hypersurface $$\Big\{(\xi_1,...,\xi_{n-1},\xi_1^m+...+\xi_{n-1}^m): (\xi_1,...,\xi_{n-1}) \in [0,1]^{n-1}\Big\}$$associated with the decomposition adapted to hypersufaces of finite type, where $n\geq 2$ and $m\geq 4$ is an even number. The key ingredients of the proof include an $\ell^2$ decoupling inequality for the hypersurfaces $$\Big\{(\xi_1,...,\xi_{n-1},\phi_1(\xi_1)+...+\phi_s(\xi_s)+\xi_{s+1}^m+...+\xi_{n-1}^m): (\xi_1,...,\xi_{n-1}) \in [0,1]^{n-1}\Big\},$$ $0 \leq s \leq n-1$, with $\phi_1,...,\phi_s$ being $m$-nondegenerate.