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 f₏, ₀, whose Newton polyhedron at infinity is a simplicial polytope P. First, we consider the stacky fan P given by P and show that for each quotient stacky fan of P, 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 f₏, ₀ 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.