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

Abstract

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.