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

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Correspondence to E. Y. S. Chan.

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Dedicated to Jonathan M. Borwein.

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Chan, E.Y.S., Corless, R.M. Minimal Height Companion Matrices for Euclid Polynomials. Math.Comput.Sci. 13, 41–56 (2019). https://doi.org/10.1007/s11786-018-0364-2

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Keywords

  • Bohemian eigenvalues
  • Minimal height
  • Companion matrix
  • Conditioning
  • Euclid numbers

Mathematics Subject Classification

  • 11C20
  • 15A22
  • 65F15
  • 65F35