Expand all Collapse all | Results 1 - 7 of 7 |
1. CMB 2012 (vol 56 pp. 759)
A Generalization of a Theorem of Boyd and Lawton The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of
$\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that
the Mahler measure of a multivariate polynomial is the limit of Mahler
measures of univariate polynomials. We prove the analogous
result for different extensions of Mahler measure such as generalized
Mahler measure (integrating the maximum of $\log|P|$ for possibly
different $P$'s),
multiple Mahler measure (involving products of $\log|P|$ for possibly
different $P$'s), and higher Mahler measure (involving $\log^k|P|$).
Keywords:Mahler measure, polynomial Categories:11R06, 11R09 |
2. CMB 2011 (vol 55 pp. 26)
A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$-Series We present another example of a $3$-variable polynomial defining a $K3$-hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$-series.
Keywords:modular Mahler measure, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, LivnÃ© criterion, Rankin-Cohen brackets Categories:11, 14D, 14J |
3. CMB 2011 (vol 54 pp. 739)
The Infimum in the Metric Mahler Measure Dubickas 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 $\alpha$ 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.
Keywords:Weil height, Mahler measure, metric Mahler measure, Lehmer's problem Categories:11R04, 11R09 |
4. CMB 2007 (vol 50 pp. 191)
Every Real Algebraic Integer Is a Difference of Two Mahler Measures We prove that every real
algebraic integer $\alpha$ is expressible by a
difference of two Mahler measures of integer polynomials.
Moreover, these polynomials can be chosen in such a way that they
both have the same degree as that of $\alpha$, say
$d$, one of these two polynomials is irreducible and
another has an irreducible factor of degree $d$, so
that $\alpha=M(P)-bM(Q)$ with irreducible polynomials
$P, Q\in \mathbb Z[X]$ of degree $d$ and a
positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.
Keywords:Mahler measures, Pisot numbers, Pell equation, $abc$-conjecture Categories:11R04, 11R06, 11R09, 11R33, 11D09 |
5. CMB 2002 (vol 45 pp. 231)
Erratum:~~The Lehmer Polynomial and Pretzel Links Erratum to {\it The Lehmer Polynomial and Pretzel Links},
Canad. J. Math. {\bf 44}(2001), 440--451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 |
6. CMB 2002 (vol 45 pp. 196)
Mahler Measures Close to an Integer We prove that the Mahler measure of an algebraic number cannot be too
close to an integer, unless we have equality. The examples of certain
Pisot numbers show that the respective inequality is sharp up to a
constant. All cases when the measure is equal to the integer are
described in terms of the minimal polynomials.
Keywords:Mahler measure, PV numbers, Salem numbers Categories:11R04, 11R06, 11R09, 11J68 |
7. CMB 2001 (vol 44 pp. 440)
The Lehmer Polynomial and Pretzel Links In this paper we find a formula for the Alexander polynomial
$\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with
$(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and
$p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7}
(x)$ is the well-known Lehmer polynomial, which is conjectured to have
the smallest Mahler measure among all monic integer polynomials. We
confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among
the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 |