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1. CMB Online first
Weak arithmetic equivalence Inspired by the invariant of a number field given by its zeta
function, we define the notion of weak arithmetic equivalence and show
that under certain ramification hypotheses, this equivalence
determines the local root numbers of the number field. This is
analogous to a result of Rohrlich on the local root numbers of a
rational elliptic curve. Additionally, we prove that for tame
non-totally real number fields, the integral trace form is invariant
under arithmetic equivalence.
Keywords:arithmeticaly equivalent number fields, root numbers Categories:11R04, 11R42 |
2. CMB 2012 (vol 56 pp. 602)
Resultants of Chebyshev Polynomials: A Short Proof We give a simple proof of the value of the resultant of two Chebyshev polynomials
(of the first or the second kind),
values lately obtained by D. P. Jacobs, M. O. Rayes and V. Trevisan.
Keywords:resultant, Chebyshev polynomials, cyclotomic polynomials Categories:11R09, 11R04 |
3. CMB 2011 (vol 54 pp. 645)
An Extension of Craig's Family of Lattices Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of
unity. The lattices in Craig's family are $(p-1)$-dimensional and
are geometrical representations of the integral $\mathbb
Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a
positive integer. This lattice construction technique is a powerful
one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$,
Craig's lattices are the densest packings known. Motivated by this,
we construct $(p-1)(q-1)$-dimensional lattices from the integral
$\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle
1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$
and $j$ are positive integers. In terms of sphere-packing density,
the new lattices and those in Craig's family have the same
asymptotic behavior. In conclusion, Craig's family is greatly
extended while preserving its sphere-packing properties.
Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields Categories:11H31, 11H55, 11H50, 11R18, 11R04 |
4. 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 |
5. 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 |
6. 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 |
7. 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 |
8. 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 |