Advertisement

# A Criterion for Polynomials to Divide Infinitely Many k- Nomials

• Lajos Hajdu
• Robert Tijdeman
Conference paper
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

11C08 (11D57)

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Evertse, J.-H., Schlickewei, H.P., Schmidt, W.M.: Linear equations in variables which lie in a multiplicative group. Ann. Math. 155, 1–30 (2002)
2. 2.
Győry, K., Schinzel, A.: On a conjecture of Posner and Rumsey. J. Number Theory 47, 63–78 (1994)
3. 3.
Hajdu, L.: On a problem of Győry and Schinzel concerning polynomials. Acta Arith. 78, 287–295 (1997)
4. 4.
Hajdu, L., Tijdeman, R.: Polynomials dividing infinitely many quadrinomials or quintinomials. Acta Arith. 107, 381–404 (2003)
5. 5.
Posner, E.C., Rumsey, H. Jr.: Polynomials that divide infinitely many trinomials. Mich. Math. J. 12, 339–348 (1965)
6. 6.
Schlickewei, H.P., Viola, C.: Polynomials that divide many trinomials. Acta Arith. 78, 267–273 (1997)
7. 7.
Schlickewei, H.R, Viola, C.: Polynomials that divide many k-nomials. In: Győry, K., Iwaniec, H., Urbanowicz, J. (eds.) Number Theory in Progress, vol. 1, pp. 445–450. de Gruyter, Berlin (1999)Google Scholar
8. 8.
Schlickewei, H.R, Viola, C.: Generalized Vandermonde determinants. Acta Arith. 95, 123–137 (2000)

## 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