Minimal height companion matrices for Euclid polynomials

TL;DR

This paper introduces minimal height companion matrices for Euclid polynomials, providing explicit constructions and exploring their properties, which could facilitate more efficient polynomial root-finding methods.

Contribution

The paper presents a novel construction of minimal height companion matrices for Euclid polynomials, advancing understanding of their algebraic structure and potential computational advantages.

Findings

Constructed explicit minimal height companion matrices for Euclid polynomials.

Proved key properties of these matrices and polynomials.

Provided experimental evidence supporting conjectured properties.

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 $E_k(\lambda) = \operatorname{det}\left(\lambda\mathbf{I} - \mathbb{E}_{k}\right)$, 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.