Geometriae Dedicata

, Volume 152, Issue 1, pp 153–164 | Cite as

Height reducing property of polynomials and self-affine tiles

Original Paper


A monic polynomial \({f(x)\in {\mathbb Z}[x]}\) is said to have the height reducing property (HRP) if there exists a polynomial \({h(x)\in {\mathbb Z}[x]}\) such that
$$f(x)h(x)=a_n x^n+a_{n-1}x^{n-1}+\cdots+a_1x\pm q,$$
where q = f(0), |a i | ≤ (|q| −1), i = 1, . . . , n and a n > 0. We show that any expanding monic polynomial f(x) has the height reducing property, improving a previous result in Kirat et al. (Discrete Comput Geom 31: 275–286, 2004) for the irreducible case. The proof relies on some techniques developed in the study of self-affine tiles. It is constructive and we formulate a simple tree structure to check for any monic polynomial f(x) to have the HRP and to find h(x). The property is used to study the connectedness of a class of self-affine tiles.


Connectedness Expanding polynomials Self-affine tiles 

Mathematics Subject Classification (2000)

28A80 11R09 


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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsCentral China Normal UniversityWuhanChina
  2. 2.Department of MathematicsIstanbul Technical UniversityMaslak, IstanbulTurkey
  3. 3.Department of MathematicsThe Chinese University of Hong KongShatin, Hong KongChina

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