The argument cycle and the coamoeba

Abstract

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 Gröbner 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 θ on the n:th torus. For this purpose, we associate a certain algebraic cycle, the argument cycle, to V and θ , and study its homology. In particular, we give a method to approximate the coamoeba when n = 2.