[Paper Review] Comments on the height reducing property

Shigeki Akiyama, Toufik Zaı̈mi|arXiv (Cornell University)|May 6, 2012

semigroups and automata theory22 references3 citations

TL;DR

This paper investigates the height reducing property for algebraic numbers, proving that complex numbers with all conjugates on the unit circle or all of modulus greater than one satisfy this property under specific multiplicative dependency conditions. It establishes that the property holds when the number of multiplicatively independent conjugates is at least deg(α)/2 − 1 or exactly 1, and provides algorithmic methods to compute these dependencies using LLL reduction, with extensive computational verification up to degree 16.

ABSTRACT

A complex number alpha is said to satisfy the height reducing property if there is a finite subset F of the ring Z of the rational integers such that Z[alpha]=F[alpha]. This problem of finding F has been considered by several authors, especially in contexts related to self affine tilings, and expansions of real numbers in non-integer bases. We continue, in this note, the description of the numbers satisfying the height reducing property, and we specify a related characterization of the roots of integer polynomials with dominant term.

Motivation & Objective

• To characterize algebraic numbers α ∈ ℂ for which ℤ[α] = F[α] for some finite subset F ⊂ ℤ, known as the height reducing property.
• To determine sufficient conditions under which algebraic numbers with conjugates on the unit circle satisfy the height reducing property.
• To develop and apply an algorithm using the LLL algorithm to compute the number of multiplicatively independent conjugates m(α), enabling verification of the height reducing property.
• To investigate the validity of the height reducing property for algebraic numbers of degree up to 16 with coefficients bounded by h ≤ 50, particularly in cases where m(α) < deg(α)/2.

Proposed method

• Define the height reducing property as ℤ[α] = F[α] for finite F ⊂ ℤ, and analyze its implications for algebraic integers and roots of unity.
• Use Kronecker’s theorem and properties of reciprocal polynomials to analyze conjugates on the unit circle.
• Introduce the concept of multiplicative independence of conjugates and define m(α) as the maximal number of multiplicatively independent conjugates.
• Apply Lemma 3.7 from de Weger [4] combined with the LLL algorithm to bound and compute m(α) efficiently, overcoming the exponential bounds from Waldschmidt [12].
• Perform systematic enumeration of monic integer polynomials of bounded degree d, leading coefficient c, and coefficient height h to search for counterexamples.
• Use sieving techniques to filter polynomials with all roots on the unit circle, and further filter for irreducibility, primitivity, and non-power-reducible forms.

Experimental results

Research questions

• RQ1 Under what conditions does an algebraic number α with all conjugates on the unit circle satisfy the height reducing property?
• RQ2 Does the height reducing property hold for all algebraic numbers of degree ≤ 6 whose conjugates lie on the unit circle?
• RQ3 Can the height reducing property be established for algebraic numbers with m(α) = deg(α)/2 − 1, deg(α)/2 − 2, or deg(α)/2 − 3?
• RQ4 Are there algebraic numbers with conjugates on the unit circle that do not satisfy the height reducing property, and if so, what is the minimal degree for such counterexamples?
• RQ5 Can the LLL algorithm be effectively used to compute m(α) and verify the height reducing property in high-degree cases?

Key findings

• The height reducing property holds for all algebraic numbers α whose conjugates are either all of modulus 1 or all greater than 1, provided m(α) ≥ deg(α)/2 − 1 or m(α) = 1.
• For all algebraic numbers of degree ≤ 6 with conjugates on the unit circle, the height reducing property holds, as m(α) ≤ deg(α)/2 − 1 is impossible.
• Among 1,030,301 polynomials of degree 6, leading coefficient 2, and coefficient height 50, 287 have all roots on the unit circle, and all satisfy the height reducing property.
• In degree 12, four polynomials were found with m(α) = deg(α)/2 − 2 and two with m(α) = deg(α)/2 − 3; the height reducing property was proven for the former via solving systems of inequalities.
• For degree 16, one polynomial with m(α) = deg(α)/2 − 3 was found, but the height reducing property remains unproven for this case.
• The search up to degree 16 found no counterexamples to the height reducing property among polynomials with conjugates on the unit circle, suggesting the property may hold universally in this class.