Comments on the height reducing property II

A complex number $\alpha$ is said to satisfy the height reducing property if there is a finite set $F\subset \mathbb{Z}$ such that $\mathbb{Z}[\alpha]=F[\alpha]$, where $\mathbb{Z}$ is the ring of the rational integers. It is easy to see that $\alpha$ is an algebraic number when it satisfies the height reducing property. We prove the relation $\operatorname{Card}(F)\geq \max\{2,\left\vert M_{\alpha}(0)\right\vert \},$ where $M_{\alpha}$ is the minimal polynomial of $\alpha$ over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers $\alpha$. In addition, we show that there is an algorithm to determine the minimal height polynomial of a given algebraic number, provided it has no conjugate of modulus one.