Comments on the height reducing property II

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.Card(F ) ≤ Card(S)(Card(S) s+1 − 1)/(Card(S) − 1), and, hence, J (t−s) (0) = β which implies that β ∈ U . Thus U ⊂ ℘, a contradiction, because by (9) we have that cZ ⊂ U and so U cannot be finite.

Characterisation of the numbers which satisfy the height reducing property

“…Proof of the theorem. The direct implication is a corollary of Theorem 1 in [2]. By iterating Lemma 1, we obtain the other implication, using Lemmas 2 and 3.…”

“…which sends α to its conjugates, over Q, situated in {z ∈ C | Im(z) ≥ 0} (for example). This allows us sometimes to obtain number systems, when α is an expanding number, but not when |α| = 1 (see for instance [2,Section 2]). An alternative solution to this problem is to add certain finite completions, corresponding to the divisors of the denominator of the fractional ideal (α), to enlarge the ring K ∞ and the range of the corresponding embedding Φ ∞ : this is the key of Lemma 1, which is the main result of this manuscript.…”

“…Following [1], we say that a complex number α satisfies the height reducing property, in short HRP, if there is a finite subset F of the ring of the rational integers Z, such that each polynomial with coefficients in Z, evaluated at α, belongs to the family F [α] :=    n j=0 ε j α j | (ε 0 , ..., ε n ) ∈ F n+1 , n ∈ N    , where N is the set of non-negative rational integers. In this case, we have, by [3, Theorem 1 (i)], that α is an algebraic number whose conjugates, over the field of the rationals Q, are all of modulus one, or all of modulus greater than one (such a number α is called an expanding number [2]). Theorem 1 (ii) of [3] says also that α satisfies HRP, when it is a root of unity, or when it is an expanding number.…”

Characterization of rational matrices that admit finite digit representations

“…Let P ∈ Z[x] be a polynomial in a single variable x with integer coefficients. Similarly to (1), for a finite subset N ⊂ Z, we define With all this in mind, we can now recast the main result of [4] in the polynomial form as follows.…”

“…Regarding Problem 10, we know that in the case where the characteristic polynomial of A is irreducible over Q and all the eigenvalues are of absolute value 1, the construction in the proof of Theorem 3 provided in [4] yields only a very rough bound for #D that is surely far from the optimum.…”