Comments on the height reducing property II

A complex number αα is said to satisfy the height reducing property if there is a finite set F⊂ZF⊂Z such that Z[α]=F[α]Z[α]=F[α], where ZZ is the ring of the rational integers. It is easy to see that αα is an algebraic number when it satisfies the height reducing property. We prove the relation Card(F)≥max{2,|Mα(0)|}Card(F)≥max{2,|Mα(0)|}, where MαMα is the minimal polynomial of αα over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers αα. 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.