Invariant factors of degree matrices and L-functions of certain exponential sums

Let f be a multivariate Laurent polynomial over a finite field and L * ( f , T ) the corresponding L-function of the toric exponential sum of f. In this paper, we obtain an explicit formula for the L-function L * ( f , T ) in terms of Gauss sums provided that the invariant factors of the degree matrix of f satisfy certain conditions. As an application, we also compute the zeta functions for some hypersurfaces.