A stratification result for an exponential sum modulo $p^2$

In this note we consider algebraic exponential sums over the values of homogeneous nonsingular polynomials $F(x_1, \cdots, x_n) \in $\mathbb{Z}$[x_1, \cdots, x_n]$inthequotientring$$\mathbb{Z}$/p^2$\mathbb{Z}$$. We provide an estimate of this exponential sum and a corresponding stratification of the space ${\mathbb{A}}_{{\mathbb{F}}_p}^n$, which in particular illustrates a general stratification theorem of Fouvry and Katz.