In this thesis we study the topology and geometry of the coamoeba of an algebraic variety V. The strategy that we use is to relate the coamoebas corresponding to the initial varieties of V to the coamoeba of V. We also define an analogue of the Ronkin function for the coamoeba and give an explicit formula for it.
We investigate the coamoeba of a complex algebraic variety V???(C*) n through the study of initial forms of the defining ideal. By use of a universal Grobner basis, we prove that the closure of the coamoeba is included in the union of coamoebas corresponding to all initial ideals. We also study complete intersections V of dimension n/2 more closely to get a lower bound for the multiplicity in V of a given point ? on the n:th torus. For this purpose, we associate a certain algebraic cycle, the argument cycle, to V and ? , and study its homology. In particular, we give a method to approximate the coamoeba when n?=?2.
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