Height reducing property of polynomials and self-affine tiles

Abstract

The authors show that for any expanding monic polynomial \(f(x)\in \mathbb{Z}[x]\) there exists a polynomial \(h(x)\in \mathbb{Z}[x]\) such that \[ f(x)h(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots + a_{1}x \pm q, \] where \(q= f(0), \mid a_{i}\mid \leq \mid q \mid -1 \; (i=1,\ldots , n)\) and \(a_{n}> 0.\) This extends an earlier result of \textit{I. Kirat} et al. [Discrete Comput. Geom. 31, No. 2, 275--286 (2004; Zbl 1054.52012)]. The proof makes use of tools from the theory of self-affine tiles.