Hasse Polynomials of L-functions of Certain Exponential Sums

Abstract

In this paper, we focus on computing the higher slope Hasse polynomials of L-functions of certain exponential sums associated to the following family of Laurent polynomials $f(x_1,\ldots ,x_{n+1})=\sum_{i=1}^na_i x_{n+1}\left(x_i+\frac{1}{x_i}\right)+a_{n+1} x_{n+1}+\frac{1}{x_{n+1}}$, where $a_i \in \F^*_{q}$, $i=1,2, \ldots, n+1$. We find a simple formula for the Hasse polynomial of the slope one side and study the irreducibility of these Hasse polynomials. We will also provide a simple form of all the higher slope Hasse polynomials for $n=3$, answering an open question of Zhang and Feng.