Height reducing property of polynomials and self-affine tiles

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.