Jens Forsgård: On Hypersurface Coamoebas and Integral Representations of A-Hypergeometric Functions

This thesis is concerned with coamoebas of hypersurfaces, and their connection to integral representations of A-hypergeometric functions. In the first part, we introduce the lopsidedness criterion for coamoebas, and define the lopsided coamoeba. We show that the set of connected components of the complement of the closed lopsided coamoeba comes naturally equipped with an order map. Using this order map we obtain new results concerning the geometry of coamoebas. In the second part, we study a class of Euler type hypergeometric integrals arising from connected components of the complement of the coamoeba, known as Euler-Mellin integrals. Through the order map for the lopsided coamoeba, we find a relation to so called Mellin-Barnes integrals. We end with a motivating example showing that Euler-Mellin integrals can be used to study the $A$-hypergeometric system also at rank jumping parameters.