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
$f(x)h(x)=a_n x^n+a_{n-1}x^{n-1}+cdots+a_1xpm q,$
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.