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A Criterion for Polynomials to Divide Infinitely Many k- Nomials

  • Lajos Hajdu
  • Robert Tijdeman
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

A polynomial Q ∈ ℚ[x] of the form
$$ Q\left( x \right) = \sum\limits_{i = 1}^k {a_i x^{m_i } } with m_1 > \ldots > m_{k - 1} > m_k = 0 and a_1 = 1 $$
is called a standard k-nomial. It is worth to mention that the restriction to monic k-nomials is only for convenience. We may replace every standard k-nomial by any of its constant multiples, and the theorems would still be valid. We call (m 1 , ..., m k ) the exponent k-tuple of Q. Note that if Q is a standard k-nomial, but not a standard (k-1)-nomial, then its exponent k-tuple is uniquely determined. Let
$$ \begin{gathered} PR_k = \left\{ {P \in \mathbb{Q}\left[ x \right]: \exists Q \in \mathbb{Q}\left[ x \right] and r \in \mathbb{Z} with deg \left( Q \right) < k} \right. \hfill \\ \left. {and r \geqslant 1 such that P\left( x \right)\left| { Q\left( {x^r } \right)over \mathbb{Q}} \right.} \right\}. \hfill \\ \end{gathered} $$

Keywords

Polynomials k-nomials quintinomials transitivity subspace theorem 

2000 Mathematics subject classification

11C08 (11D57) 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Lajos Hajdu
    • 1
    • 2
  • Robert Tijdeman
    • 3
  1. 1.Number Theory Research GroupHungarian Academy of SciencesDebrecenHungary
  2. 2.Institute of MathematicsUniversity of DebrecenDebrecenHungary
  3. 3.Mathematical InstituteLeiden UniversityLeidenThe Netherlands

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