Minimal height companion matrices for Euclid polynomials

Abstract

We define Euclid polynomials $E_{k+1}(\lambda )=E_k(\lambda )\left(E_k(\lambda )-1\right)+1$ and $E_1(\lambda )=\lambda +1$ in analogy to Euclid numbers $e_k=E_k(1)$. We show how to construct companion matrices ${\mathbb{E}}_k$, so $E_k(\lambda )=\det \left(\lambda \mathbf{I}-{\mathbb{E}}_k\right)$, of height 1 (and thus of minimal height over all integer companion matrices for $E_k(\lambda )$). We prove various properties of these objects, and give experimental confirmation of some unproved properties.

15 pages, 7 figures

References in corpus (2)

• Unimodal sequences show Lambert W is Bernstein
• The polynomial eigenvalue problem is well conditioned for random inputs

Cited by in corpus (1)

• Bohemian Upper Hessenberg Matrices