Abstract
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
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.
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The research is partially supported by CUHK Focus Investment Schems and the National Natural Science Foundation of China 10771082 and 10871180.
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He, XG., Kirat, I. & Lau, KS. Height reducing property of polynomials and self-affine tiles. Geom Dedicata 152, 153–164 (2011). https://doi.org/10.1007/s10711-010-9550-3
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DOI: https://doi.org/10.1007/s10711-010-9550-3