Height reducing property of polynomials and self-affine tiles

A monic polynomial f(x)ε ℤ[x] is said to have the height reducing property (HRP) if there exists a polynomial f(x)ε ℤ[x] such that where q = f(0), {Pipe}a_i{Pipe} ≤ ({Pipe}q{Pipe} -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.