# Amoebas, Discriminants, and Hypergeometric Functions

Abstract: This thesis consists of six chapters.In Chapter 1 we give some historical background to the topic of the thesis together with the fundamental definitions and results that the thesis is based on.In Chapter 2 we study Mellin transforms of rational functions and investigate their analytic continuations. The main result in this chapter is a full description of the polar locusof the meromorphic continuation of the Mellin transform. It turns out tobe closely connected with the Newton polytope of the denominator f of the rational function. We also relate the Mellin transforms to the coamoeba of the polynomial f. In fact, we represent the function 1/f as an inverse Mellin transform converging on the complement of the coamoeba of f. This is in analogy with the Laurent series expansions of 1/f which are known to converge on the complement of the amoeba of f.In Chapter 3 we study the general structure and properties of two dimensional discriminantal coamoebas. We prove that such a coamoeba is the union of two mirror images of a polygonal curve simply obtained from the matrix B in the Horn-Kapranov parametrization. We provide an area formula for the coamoeba, and show that the coamoeba is intimately related to acertain zonotope. In fact, considering the coamoeba and the zonotope as chains projected on the torus (R/2piZ)^2, the summed chain obtained as the union of the coamoeba and the zonotope is a 2-cycle, and as such, is an integer multiple of the torus itself.The last three chapters deal with hypergeometric functions, again in connection with amoeba theory. We study A-hypergeometric functions in the form of power series, and analytic continuations given by integrals of Mellin-Barnes type. We also introduce a related Gamma-integral, which is more suitable as a continuous version of the Gamma-series. We prove the orems describing the domains of convergence forA-hypergeometric series and for the associated Mellin-Barnes typeintegrals, as well as for the Gamma-integrals. The exact description of the convergence domains is given in terms of the complement components of discriminantal amoebas for the series, whereas in the case of the integrals they are given as zonotopes. By the results in Chapter 3, we know (for two dimensions) that these zonotopes exactly cover the complement of the coamoeba the correct number of times in order to get acomplete basis of hypergeometric integrals.

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