Comments on the height reducing property II

A complex number $α$ is said to satisfy the height reducing property if there is a finite set $F\subset \mathbb{Z}$ such that $\mathbb{Z}[α]=F[α]$, where $\mathbb{Z}$ is the ring of the rational integers. It is easy to see that $α$ is an algebraic number when it satisfies the height reducing property. We prove the relation $\operatorname{Card}(F)\geq \max\{2,\left\vert M_α(0)\right\vert \},$ where $M_α$ is the minimal polynomial of $α$ over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers $α$. 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.

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