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.
Categories: math.AP