Newton polygons for $L$-functions of generalized Kloosterman sums

In this paper, we study the Newton polygons for the $L$-functions of $n$-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top dimensional Dwork cohomology. Using Wan's decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for $L$-function of $\bar{F}(\bar{\lambda},x):=\sum_{i=1}^nx_i^{a_i}+\bar{\lambda}\prod_{i=1}^nx_i^{-1}$.