Disproving a conjecture of Thornton on Bohemian matrices

Zhibin Du,Jiahao Ye,Carlos M Da Fonseca,Yingqiu Xu

doi:10.1515/math-2021-0045

Abstract

Abstract In this paper, we disprove a remaining conjecture about Bohemian matrices, in which the numbers of distinct determinants of a normalized Bohemian upper-Hessenberg matrix were conjectured.

Highlights

A matrix is called Bohemian if its entries are of a bounded height, typically drawn from a discrete set
We find the possible intervals of det A, for any A ∈ n, up to n = 8
In a matrix A = of order 3, the unknown entries are a12, a13, a23 listed in the lexicographic ordering, we would assign a12 ≔ A[0], a13 ≔ A[1], a23 ≔ A[2] according to Algorithm 1

Summary

Introduction

A matrix is called Bohemian if its entries are of a bounded height, typically drawn from a discrete set. The term “Bohemian” is intended as a mnemonic and derived from “BOunded HEight Matrix of Integers.”. These matrices are of relevant interest in many areas where their spectral properties are relevant. The most important collection can be found in the so-called Characteristic Polynomial Database [6]. Quite recently, this catalogue attracted the attention of several researchers, namely on those conjectures concerned to Hessenberg matrices. This catalogue attracted the attention of several researchers, namely on those conjectures concerned to Hessenberg matrices They are extended and proved in [7,8,9]. We recall that the first 11 terms of sequence [11, A212264] are listed in Table 1, where we understand A212264n as the nth term of the sequence A212264

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Matrices in n

The distribution of determinants for matrices in n

Concluding remarks