scholarly journals Lehmer's Conjecture for Hermitian Matrices over the Eisenstein and Gaussian Integers

10.37236/2834 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Graeme Taylor ◽  
Gary Greaves
Keyword(s):  
Mahler Measure ◽  
Hermitian Matrices ◽  
Gaussian Integers ◽  
Lehmer's Conjecture ◽  
Lehmer’S Problem

We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer's number $\tau_0 = 1.17628\dots$.

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 A SHORT NOTE ON THE MAHLER MEASURE AND LEHMER'S CONJECTURE

2017 ◽  
Vol 17 (1) ◽  
pp. 47-60
Author(s):  
Ali H. Hakami
Keyword(s):  
Short Note ◽  
Mahler Measure ◽  
Lehmer's Conjecture

CONGRUENCES AND LEHMER'S PROBLEM

2008 ◽  
Vol 04 (04) ◽  
pp. 587-596 ◽  
Author(s):  
NICOLAE CIPRIAN BONCIOCAT
Keyword(s):  
Lower Bounds ◽  
Mahler Measure ◽  
Integer Coefficients ◽  
Lehmer’S Problem

We obtain explicit lower bounds for the Mahler measure for nonreciprocal polynomials with integer coefficients satisfying certain congruences.

scholarly journals HEDGEHOGS IN LEHMER’S PROBLEM

2021 ◽  
pp. 1-7
Author(s):  
JAN-WILLEM M. VAN ITTERSUM ◽  
BEREND RINGELING ◽  
WADIM ZUDILIN
Keyword(s):  
Mahler Measure ◽  
Lehmer’S Problem

Abstract Motivated by a famous question of Lehmer about the Mahler measure, we study and solve its analytic analogue.

An algorithm for eigenvectors of non-hermitian matrices

1969 ◽  
Author(s):  
Albert Maurice Erisman
Keyword(s):  
Hermitian Matrices

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

The Mahler measure of a three-variable family and an application to the Boyd–Lawton formula

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Jarry Gu ◽  
Matilde Lalín
Keyword(s):  
Mahler Measure

Voronoi summation formula for Gaussian integers

2021 ◽  
Author(s):  
Debika Banerjee ◽  
Ehud Moshe Baruch ◽  
Daniel Bump
Keyword(s):  
Summation Formula ◽  
Gaussian Integers ◽  
Voronoi Summation

scholarly journals Eigenbranes in Jackiw-Teitelboim gravity

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Andreas Blommaert ◽  
Thomas G. Mertens ◽  
Henri Verschelde
Keyword(s):  
Finite Volume ◽  
Chaotic System ◽  
Path Integral ◽  
Hermitian Matrices ◽  
The Matrix ◽  
Integral Contour

Abstract It was proven recently that JT gravity can be defined as an ensemble of L × L Hermitian matrices. We point out that the eigenvalues of the matrix correspond in JT gravity to FZZT-type boundaries on which spacetimes can end. We then investigate an ensemble of matrices with 1 ≪ N ≪ L eigenvalues held fixed. This corresponds to a version of JT gravity which includes N FZZT type boundaries in the path integral contour and which is found to emulate a discrete quantum chaotic system. In particular this version of JT gravity can capture the behavior of finite-volume holographic correlators at late times, including erratic oscillations.

scholarly journals A cellular basis for the generalized Temperley–Lieb algebra and Mahler measure

2014 ◽  
Vol 178 ◽  
pp. 107-124
Author(s):  
Xuanting Cai ◽  
Robert G. Todd
Keyword(s):  
Mahler Measure ◽  
Cellular Basis
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