Decoupling for finite type phases in higher dimensions

TL;DR

This paper proves an $ ext{l}^2$ decoupling inequality for a specific class of hypersurfaces of finite type in higher dimensions, extending decoupling techniques to more complex geometric structures.

Contribution

It establishes a new $ ext{l}^2$ decoupling inequality for hypersurfaces of finite type, including those with mixed polynomial and nondegenerate components, in higher dimensions.

Findings

Proved $ ext{l}^2$ decoupling inequality for the given hypersurface.

Extended decoupling methods to hypersurfaces with finite type and mixed structures.

Provided key ingredients involving decoupling for hypersurfaces with nondegenerate components.

Abstract

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.