Reciprocal Algebraic Integers Whose Mahler Measures are Non-Reciprocal

1987 ◽  
Vol 30 (1) ◽  
pp. 3-8 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Unit Circle ◽  
Algebraic Integer ◽  
Mahler Measure ◽  
Algebraic Integers

AbstractThe Mahler measure M (α) of an algebraic integer α is the product of the moduli of the conjugates of α which lie outside the unit circle. A number α is reciprocal if α- 1 is a conjugate of α. We give two constructions of reciprocal a for which M (α) is non-reciprocal producing examples of any degree n of the form 2h with h odd and h ≥ 3, or else of the form with s ≥ 2. We give explicit examples of degrees 10, 14 and 20.

scholarly journals Perron units which are not Mahler measures

1986 ◽  
Vol 6 (4) ◽  
pp. 485-488 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Unit Circle ◽  
Ergodic Theory ◽  
Minimal Polynomial ◽  
Algebraic Integer ◽  
Mahler Measure ◽  
Absolute Value ◽  
Perron Number ◽  
The Absolute

AbstractThe Mahler measure M(α) of an algebraic integer α is the product of the absolute value of the conjugates of α which lie outside the unit circle. The quantity log M(α) occurs in ergodic theory as the entropy of an endomorphism of the torus. Adler and Marcus showed that if β = M(α) then β is a Perron number which is a unit if α is a unit. They asked whether the Perron number β whose minimal polynomial is tm −t −1 is the measure of any algebraic integer. We show here that the answer is negative for all m > 3.

On ℤ-Modules of Algebraic Integers

2009 ◽  
Vol 61 (2) ◽  
pp. 264-281 ◽  
Author(s):  
J. P. Bell ◽  
K. G. Hare
Keyword(s):  
Algebraic Integer ◽  
Full Rank ◽  
Pisot Number ◽  
List Type ◽  
Algebraic Integers ◽  
Pisot Numbers ◽  
Cyclotomic Numbers ◽  
Special Cases ◽  
Fixed Order ◽  
Fixed Positive Integer

Abstract. Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of ℂ* generated by the conjugates of q. We say q is of full rank if either the rank is d − 1 and q has norm ±1, or the rank is d. In this paper we study some properties of ℤ[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results.(1)If q is an algebraic integer of full rank and n is a fixed positive integer, then there are only finitely many m such that disc `ℤ[qm]´ = disc `ℤ[qn]´.(2)If q and r are algebraic integers of degree d of full rank and ℤ[qn] = ℤ[rn] for infinitely many n, then either q = ωr′ or q = Norm(r)2/dω/r′ , where r ′ is some conjugate of r and ω is some root of unity.(3)Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that ℤ[q] = ℤ[r].(4)There are only finitely many Pisot-cyclotomic numbers of any fixed order.

scholarly journals On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

2016 ◽  
Vol 11 (1) ◽  
pp. 79-139 ◽  
Author(s):  
Jean-Louis Verger-Gaugry
Keyword(s):  
Unit Circle ◽  
Asymptotic Expansions ◽  
Direct Method ◽  
Direct Proof ◽  
Formal Series ◽  
Mahler Measure ◽  
Pisot Number ◽  
Number Comparison ◽  
Lehmer’S Problem ◽  
Series Of Functions

AbstractLet n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers $(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1. We prove that the Conjecture of Lehmer is true for $\{ \theta _n^{ - 1} |n \ge 2\} $ by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn, zj,n, of Gn(X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures ${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$ of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for $\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house , and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots of Gn, near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.

scholarly journals On the Zhang–Zagier measure

2018 ◽  
Vol 14 (10) ◽  
pp. 2663-2671
Author(s):  
V. Flammang
Keyword(s):  
Lower Bound ◽  
Mahler Measure ◽  
Algebraic Integers ◽  
Totally Positive ◽  
The Absolute

We first improve the known lower bound for the absolute Zhang–Zagier measure in the general case. Then we restrict our study to totally positive algebraic integers. In this case, we are able to find six points for the related spectrum. At last, we give inequalities involving the Zhang–Zagier measure, the Mahler measure and the length of such integers.

scholarly journals The Zeta Mahler measure of $(z^n − 1)/(z − 1)$

2019 ◽  
Author(s):  
Arunabha Biswas ◽  
M Ram Murty
Keyword(s):  
Unit Circle ◽  
Generating Function ◽  
Laurent Polynomial ◽  
Mahler Measure ◽  
International Audience

International audience We consider the k-higher Mahler measure $m_k (P) $ of a Laurent polynomial $P$ as the integral of ${\log}^k |P | $ over the complex unit circle and zeta Mahler measure as the generating function of the sequence ${m_k (P)}$. In this paper we derive a few properties of the zeta Mahler measure of the polynomial $P_n (z) := (z^N − 1)/(z − 1) $ and propose a conjecture.

scholarly journals Algebraic integers near the unit circle

1971 ◽  
Vol 18 ◽  
pp. 355-369 ◽  
Author(s):  
P. Blanksby ◽  
H. Montgomery
Keyword(s):  
Unit Circle ◽  
Algebraic Integers

scholarly journals Algebraic integers on the unit circle

2006 ◽  
Vol 118 (2) ◽  
pp. 189-191 ◽  
Author(s):  
Ryan C. Daileda
Keyword(s):  
Unit Circle ◽  
Algebraic Integers
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