Advertisement

Composite factors of binomials and linear systems in roots of unity

  • Roberto Dvornicich
  • Umberto Zannier
Article
  • 3 Downloads

Abstract

In this paper we completely classify binomials in one variable which have a nontrivial factor which is composite, i.e., of the shape g(h(x)) for polynomials g, h both of degree > 1. In particular, we prove that, if a binomial has such a composite factor, then deg g ≤ 2 (under natural necessary conditions). This is best-possible and improves on a previous bound deg g ≤ 24.

This result provides evidence toward a conjecture predicting a similar bound when binomials are replaced by polynomials with any given number of terms.

As an auxiliary result, which could have other applications, we completely classify the solutions in roots of unity of certain systems of linear equations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. Beukers and C. Smyth, Cyclotomic points on curves, in: Number Theory for the Millennium, I (Urbana, IL, 2000), A. K. Peters, Natick, MA, 2002, pp. 67–85.zbMATHGoogle Scholar
  2. [2]
    J. H. Conway and A. J. Jones, Trigonometric Diophantine equations (On vanishing sums of roots of unity), Acta Arithmetica 30 (1976), 229–240.MathSciNetCrossRefGoogle Scholar
  3. [3]
    R. Dvornicich and U. Zannier, On Sums of Roots of Unity, Monatshefte für Mathematk 129 (2000), 97–108.MathSciNetCrossRefGoogle Scholar
  4. [4]
    C. Fuchs and U. Zannier, Composite rational functions expressible with few terms, Journal of the European Mathematical Society 14 (2012), 175–208.MathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Schinzel, Polynomials with Special Regard to Reducibility, Encyclopedia of Mathematics and its Applications, Vol. 77, Cambridge University Press, Cambridge, 2000.Google Scholar
  6. [6]
    U. Zannier, On the number of terms of a composite polynomial, Acta Arithmetica 127 (2007), 157–167; Addendum, ibid. 140 (2009), 93–99.MathSciNetCrossRefGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItalia
  2. 2.Scuola Normale SuperiorePisaItalia

Personalised recommendations