Monodromies at Infinity of Polynomial Maps and $A$-hypergeometric Functions

Abstract

We review our recent results on monodromies at infinity of polynomial maps and $A$-hypergeometric functions. By using the theory of mixed Hodge modules, we introduce motivic global Milnor fibers of polynomial maps which encode the information of their monodromies at infinity into mixed Hodge structures with finite group actions. The numbers of the Jordan blocks in the monodromy at infinity of the polynomial will be described by its Newton polyhedron at infinity