Maximum Absolute Determinants of Upper Hessenberg Bohemian Matrices

TL;DR

This paper determines the maximum absolute determinant of a specific class of integer matrices called upper Hessenberg Bohemian matrices with fixed subdiagonal entries, extending previous results and confirming a recent conjecture.

Contribution

It extends the known results for maximum determinants of Bohemian matrices to the upper Hessenberg case with fixed subdiagonal entries and generalizes to non-integer entries.

Findings

Maximum absolute determinants for upper Hessenberg Bohemian matrices are established.

The results confirm a recent conjecture by Fasi & Negri Porzio.

The study extends to matrices with non-integer entries.

Abstract

A matrix is called Bohemian if its entries are sampled from a finite set of integers. We determine the maximum absolute determinant of upper Hessenberg Bohemian Matrices for which the subdiagonal entries are fixed to be $1$ and upper triangular entries are sampled from $\{0,1,\cdots,n\}$, extending previous results for $n=1$ and $n=2$ and proving a recent conjecture of Fasi & Negri Porzio [8]. Furthermore, we generalize the problem to non-integer-valued entries.