Generalized amoebas for subvarieties of $GL_n(\mathbb{C})$

This paper is a report based on the results obtained during a three months internship at the University of Pittsburgh by the first author and under the mentorship of the second author. The notion of an amoeba of a subvariety in a torus $(\mathbb{C}^*)^n$ has been extended to subvarieties of the general linear group $GL_n(\mathbb{C})$ by the second author and Manon. In this paper, we show some basic properties of these matrix amoebas, e.g. any such amoeba is closed and the connected components of its complement are convex when the variety is a hypersurface. We also extend the notion of Ronkin function to this setting. For hypersurfaces, we show how to describe the asymptotic directions of the matrix amoebas using a notion of Newton polytope. Finally, we partially extend the classical statement that the amoebas converge to the tropical variety. We also discuss a few examples. Our matrix amoeba should be considered as the Archimedean version of the spherical tropicalization of Tevelev-Vogiannou for the variety $GL_n(\mathbb{C})$ regarded as a spherical homogeneous space for the left-right action of $GL_n(\mathbb{C}) \times GL_n(\mathbb{C})$.