On Stein's ``Harmonic Analysis Real-variable methods, orthogonality, and oscillatory integrals'' (5.13, page 363) there is the following statement. Let $\phi$ be a real homogeneous polynomial on $\mathbb{R}^n$ of degree $k \geq 2$ that is non-degenerate, in the sense that $\det \left( \frac{\partial^2 \phi}{\partial x_i \partial x_j} \right) \not = 0$ whenever $\mathbf{x} \not = 0$. Then if $\psi \in C_0^{\infty}$, $$ \int_{\mathbb{R}^n} e^{i (\lambda \phi(\mathbf{x}) + \boldsymbol{\xi} \cdot \mathbf{x} )} \psi(\mathbf{x}) d \mathbf{x} = O((|\lambda| + |\boldsymbol{\xi}|)^{-n/k}). $$

The statement is without proof and I was wondering about how I can prove this. As I am not too familiar with this area, I was wondering if someone could provide a proof or any assistance would be appreciated. Thank you very much.