The phase limit set of linear spaces and discriminants

Frank Sottile

As an amoeba is the set of lengths  of points in a variety, its coamoeba is
the set of angles (arguments) of its points.  The set of limiting arguments
forms its  phase limit  set.  This combinatorial  backbone of  the coamoeba
reflects  the structure  of  the  tropical variety.   We  give a  recursive
description of the phase limit  set of a linear space/hyperplane complement
in terms of the flats of the  hyperplane arrangement.  We use this to study
the phase limit set  of a reduced discriminant, showing that  it is a union
of prisms  over discriminants  of lower dimension.   We conjecture  that in
dimension at least 3  the discriminant is a subset of  its phase limit set,
which implies that that coamoeba discriminant has a polyhedral structure.