scholarly journals The Mahler measure of algebraic numbers: a survey

2010 ◽  
pp. 322-349 ◽  
Author(s):  
Chris Smyth
Keyword(s):  
Mahler Measure ◽  
Algebraic Numbers

scholarly journals Mahler measure on Galois extensions

2018 ◽  
Vol 14 (06) ◽  
pp. 1605-1617 ◽  
Author(s):  
Francesco Amoroso
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Galois Extension ◽  
Mahler Measure ◽  
Algebraic Numbers ◽  
Galois Extensions

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.

scholarly journals ON THE REMAK HEIGHT, THE MAHLER MEASURE AND CONJUGATE SETS OF ALGEBRAIC NUMBERS LYING ON TWO CIRCLES

2001 ◽  
Vol 44 (1) ◽  
pp. 1-17 ◽  
Author(s):  
A. Dubickas ◽  
C. J. Smyth
Keyword(s):  
Lower Bounds ◽  
Height Function ◽  
Complete Characterization ◽  
Mahler Measure ◽  
The Other ◽  
Upper And Lower Bounds ◽  
Algebraic Numbers ◽  
Salem Numbers ◽  
Primary 11R06

AbstractWe define a new height function $\mathcal{R}(\alpha)$, the Remak height of an algebraic number $\alpha$. We give sharp upper and lower bounds for $\mathcal{R}(\alpha)$ in terms of the classical Mahler measure $M(\alpha)$. Study of when one of these bounds is exact leads us to consideration of conjugate sets of algebraic numbers of norm $\pm 1$ lying on two circles centred at 0. We give a complete characterization of such conjugate sets. They turn out to be of two types: one related to certain cubic algebraic numbers, and the other related to a non-integer generalization of Salem numbers which we call extended Salem numbers.AMS 2000 Mathematics subject classification: Primary 11R06

scholarly journals Mahler Measures as Linear Combinations of L–values of Multiple Modular Forms

2015 ◽  
Vol 67 (2) ◽  
pp. 424-449 ◽  
Author(s):  
Detchat Samart
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Mahler Measure ◽  
Laurent Polynomials ◽  
Algebraic Numbers ◽  
Quadratic Character ◽  
Linear Combinations ◽  
Special Values

AbstractWe study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves L–values of at most one newform and/or at most one quadratic character. In this paper we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to L–values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular L–values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of L–functions.

scholarly journals On algebraic numbers close to 1

1998 ◽  
Vol 58 (3) ◽  
pp. 423-434 ◽  
Author(s):  
Artūras Dubickas
Keyword(s):  
Explicit Construction ◽  
Mahler Measure ◽  
Algebraic Numbers ◽  
Weil Height ◽  
Small Height

We prove that there exists a polynomial of small height with a root close to 1. This implies that there are algebraic numbers close to 1 with relatively small Mahler measure. We also give an explicit construction of such numbers with small Weil height.

scholarly journals The Infimum in the Metric Mahler Measure

2011 ◽  
Vol 54 (4) ◽  
pp. 739-747 ◽  
Author(s):  
Charles L. Samuels
Keyword(s):  
Algebraic Number ◽  
Multiplicative Group ◽  
Mahler Measure ◽  
Algebraic Numbers ◽  
Metric Mahler Measure

AbstractDubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number α by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

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