The bohemian eigenvalue project

Abstract

Bohemian eigenvalues are the eigenvalues of matrices with entries of bounded height, typically drawn from a discrete set. We will call this set F with cardinality #F. The name "Bohemian" is intended as a mnemonic and is derived from "bounded height integer matrices." These objects are surprisingly interesting to study, with many unsolved problems related to them, and with many applications. See the works of Tao and Vu [10] for universality results for larger dimension in the generic structured case, for instance. This project concentrates on explicit construction of high resolution pictures of the eigenvalues for modest dimensions and sizes of the entries; for instance, Figure 1a is a picture of the eigenvalues of all 5 × 5 matrices with entries in {−1, 0, 1} colored by density and plotted on the complex plane.