L-functions of exponential sums and Hasse polynomials

Abstract

Let f be a non-degenerate Laurent polynomial over a finite field and L⁎(f,T) the associated L-function of the toric exponential sums of f. The Newton polygon can be used to study the p-adic valuations of roots or poles of L⁎(f,T), which has a lower bound called the Hodge polygon that depends only on the Newton polytope of f. These two polygons coincide when the coefficients of f are not zeros of the certain Hasse polynomial. Using the Dwork trace formula, we find a formula for the Hasse polynomial of the slope one side for a class of Laurent polynomials, which generalizes the results of Zhang, Feng and Chen.