Newton Polyhedrons and Hodge Numbers of Non-degenerate Laurent Polynomials

Claude Sabbah has defined the Fourier transform G of the Gauss-Manin system for a non-degenerate and convenient Laurent polynomial and has shown that there exists a polarized mixed Hodge structure on the vanishing cycle of G. In this article, we consider certain non-degenerate and convenient Laurent polynomials fP,a, whose Newton polyhedron at infinity is a simplicial polytope P. First, we consider the stacky fan ΣP given by P and show that for each quotient stacky fan of ΣP, there is a natural polarized mixed Hodge structure on the ring of conewise polynomial functions on it. Then, we describe the polarized mixed Hodge structure on the vanishing cycle associated to fP,a using these rings of conewise polynomial functions. In particular, we compute the Hodge diamond of the vanishing cycle. As a further consequence, we can solve the Birkhoff problem of such a Laurent polynomial by using elementary methods.