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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 $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ in the quotient ring $\mathbb{Z}/p^2\mathbb{Z}$. 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.