The Challenge of the Characteristic height of Bohemian Matrices: experiments with Maple and open problems

A family of Bohemian matrices is a set of matrices where the entries are independently sampled from a finite set, usually integers, of bounded height. Such families arise in many applications (e.g. compressed sensing) and the properties of matrices selected "at random" from such families are of practical and mathematical interest. Studying such matrices leads to many unanswered questions. In this paper, we state a new problem arising from the results obtained in a previous work concerning the maximal height of the characteristic polynomials of this family. We present a challenge with a computational flavour. More precisely, we describe, and illustrate by means of some experiments—executed with Maple—the problem of analyzing the behaviour of the set of heights, not only the maximal, of the characteristic polynomials of a family of upper Hessenberg Toeplitz structured Bohemian matrices.