Minimal height companion matrices for Euclid polynomials

Abstract

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