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)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)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.