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Minimal Height Companion Matrices for Euclid Polynomials

  • E. Y. S. Chan
  • R. M. Corless
Article

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

Keywords

Bohemian eigenvalues Minimal height Companion matrix Conditioning Euclid numbers 

Mathematics Subject Classification

11C20 15A22 65F15 65F35 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Ontario Research Center for Computer Algebra and School of Mathematical and Statistical Sciences, Rotman Institute of PhilosophyWestern UniversityLondonCanada

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