Abstract
A complex number α is said to satisfy the height reducing property if there is a finite set F⊂Z such that Z[α]=F[α], where 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 Card(F)≥max{2,|Mα(0)|}, where Mα 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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