Generic Newton polygon for exponential sums in $n$ variables with parallelotope base

Let $p$ be a prime number. Every $n$-variable polynomial $f(\underset{‾}{x})$ over a finite field of characteristic $p$ defines an Artin–Schreier–Witt tower of varieties whose Galois group is isomorphic to ${\mathbb{Z}}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a finite character of ${\mathbb{Z}}_p$ and a generic polynomial whose convex hull is an $n$-dimensional paralleltope $\mathrm{\Delta }$. We denote this polygon by $\mathrm{G}\mathrm{N}\mathrm{P}(\mathrm{\Delta })$. We prove a lower bound of $\mathrm{G}\mathrm{N}\mathrm{P}(\mathrm{\Delta })$, which is called the improved Hodge polygon $\mathrm{I}\mathrm{H}\mathrm{P}(\mathrm{\Delta })$. We show that $\mathrm{I}\mathrm{H}\mathrm{P}(\mathrm{\Delta })$ lies above the usual Hodge polygon $\mathrm{H}\mathrm{P}(\mathrm{\Delta })$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $\mathrm{\Delta }$, it coincides with $\mathrm{G}\mathrm{N}\mathrm{P}(\mathrm{\Delta })$ at these points. As a corollary, we roughly determine the distribution of the slopes of $\mathrm{G}\mathrm{N}\mathrm{P}(\mathrm{\Delta })$.