On the Distance Between Two Algebraic Numbers

Abstract

In this paper we give an estimate for the difference between the moduli of two roots of a polynomial with integer coefficients in terms of its degree and Mahler measure. An application of this estimate implies a stronger version of a recent result of Gómez Ruiz and Luca. In passing, we prove an estimate for the distance between two algebraic numbers in terms of their degrees and Mahler measures. For possible applications all our results are given with explicit constants. They are stated without any extra conditions or unnecessary assumptions whenever possible.

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Acknowledgements

I thank both referees for pointing out some small errors, misprints, bad notation, misspelled names, etc. This research was funded by the European Social Fund according to the activity ‘Improvement of researchers’ qualification by implementing world-class R&D projects’ of Measure No. 09.3.3-LMT-K-712-01-0037.

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Correspondence to Artūras Dubickas.

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Emrah Kilic.

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Dubickas, A. On the Distance Between Two Algebraic Numbers. Bull. Malays. Math. Sci. Soc. 43, 3049–3064 (2020). https://doi.org/10.1007/s40840-019-00855-0

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Keywords

  • Polynomial root separation
  • Mahler measure
  • Confluent Vandermonde determinant

Mathematics Subject Classification

  • 11C08
  • 11R09
  • 12D10