Comments on the height reducing property

Shigeki Akiyama; Toufik Zaimi

Open Mathematics (2013)

  • Volume: 11, Issue: 9, page 1616-1627
  • ISSN: 2391-5455

Abstract

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A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.

How to cite

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Shigeki Akiyama, and Toufik Zaimi. "Comments on the height reducing property." Open Mathematics 11.9 (2013): 1616-1627. <http://eudml.org/doc/269056>.

@article{ShigekiAkiyama2013,
abstract = {A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.},
author = {Shigeki Akiyama, Toufik Zaimi},
journal = {Open Mathematics},
keywords = {Roots of polynomials; Height of polynomials; Special algebraic numbers; Quantitative Kronecker’s approximation theorem; roots of polynomials; height of polynomials; special algebraic numbers; quantitative Kronecker's approximation theorem},
language = {eng},
number = {9},
pages = {1616-1627},
title = {Comments on the height reducing property},
url = {http://eudml.org/doc/269056},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Shigeki Akiyama
AU - Toufik Zaimi
TI - Comments on the height reducing property
JO - Open Mathematics
PY - 2013
VL - 11
IS - 9
SP - 1616
EP - 1627
AB - A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.
LA - eng
KW - Roots of polynomials; Height of polynomials; Special algebraic numbers; Quantitative Kronecker’s approximation theorem; roots of polynomials; height of polynomials; special algebraic numbers; quantitative Kronecker's approximation theorem
UR - http://eudml.org/doc/269056
ER -

References

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  9. [9] Lenstra A.K., Lenstra H.W. Jr., Lovász L., Factoring polynomials with rational coefficients, Math. Ann., 1982, 261(4), 515–534 http://dx.doi.org/10.1007/BF01457454 Zbl0488.12001
  10. [10] Mahler K., A remark on Kronecker’s theorem, Enseignement Math., 1966, 12, 183–189 Zbl0154.04702
  11. [11] Narkiewicz W., Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer Monogr. Math., Springer, Berlin, 2004 http://dx.doi.org/10.1007/978-3-662-07001-7 
  12. [12] Vorselen T., On Kronecker’s Theorem over the Adéles, MSc thesis, Universiteit Leiden, 2010, preprint available at http://www.math.leidenuniv.nl/scripties/VorselenMaster.pdf 
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