22 research outputs found
Salem-Boyd sequences and Hopf plumbing
Given a fibered link, consider the characteristic polynomial of the monodromy
restricted to first homology. This generalizes the notion of the Alexander
polynomial of a knot. We define a construction, called iterated plumbing, to
create a sequence of fibered links from a given one. The resulting sequence of
characteristic polynomials has the same form as those arising in work of Salem
and Boyd in their study of distributions of Salem and P-V numbers. From this we
deduce information about the asymptotic behavior of the large roots of the
generalized Alexander polynomials, and define a new poset structure for Salem
fibered links.Comment: 18 pages, 6 figures, to appear in Osaka J. Mat
FINDING NEW LIMIT POINTS OF MAHLER MEASURE BY METHODS OF MISSING DATA RESTORATION
It is well known that the set of Mahler measures of single variable polynomial has limit points of which a list established by D. Boyd and M. Mossinghoff has been extended through approaches based on genetic algorithms. In this paper, we wish to further extend the list of known limit points by adapting a method of missing data restoration
On Mahler's inequality and small integral generators of totally complex number fields
We improve Mahler's lower bound for the Mahler measure in terms of the
discriminant and degree for a specific class of polynomials: complex monic
polynomials of degree such that all roots with modulus greater than
some fixed value occur in equal modulus pairs. We improve Mahler's
exponent on the discriminant to . Moreover, we
show that this value is sharp, even when restricting to minimal polynomials of
integral generators of a fixed not totally real number field.
An immediate consequence of this new lower bound is an improved lower bound
for integral generators of number fields, generalising a simple observation of
Ruppert from imaginary quadratic to totally complex number fields of arbitrary
degree.Comment: To appear in Acta Arithmetic
Salem numbers and arithmetic hyperbolic groups
In this paper we prove that there is a direct relationship between Salem
numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic
groups that are determined by a quadratic form over a totally real number
field. As an application we determine a sharp lower bound for the length of a
closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each
dimension n. We also discuss a "short geodesic conjecture", and prove its
equivalence with "Lehmer's conjecture" for Salem numbers.Comment: The exposition in version 3 is more compact; this shortens the paper:
26 pages now instead of 37. A discussion on Lehmer's problem has been added
in Section 1.2. Final version, to appear is Trans. AM
The parametrized family of metric Mahler measures
AbstractLet M(α) denote the (logarithmic) Mahler measure of the algebraic number α. Dubickas and Smyth, and later Fili and the author, examined metric versions of M. The author generalized these constructions in order to associate, to each point in t∈(0,∞], a metric version Mt of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions Mt, using them to present an equivalent form of Lehmerʼs conjecture. We show that the function t↦Mt(α)t is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph t↦Mt(α) for rational α
Fields with the Bogomolov property
A field is said to have the Bogomolov property relative to a height function h’ if and only if h’ is either zero or bounded from below by a positive constant for all for all elements in this field. In this thesis we study this property according to canonical heights associated to rational functions introduced by Call and Silverman in 1994. In the first part we will translate known results into the dynamical setting. Then we prove an effective version of a theorem of Baker which states that the Néron-Tate height of an elliptic curve with multiplicative reduction at a finite place v is bounded from below by a positive constant at points which are unramified over v. In the last section of this thesis we give a complete classification of rational functions f defined over the algebraic numbers such that the maximal totally real field has the Bogomolov property relative to the canonical height associated to f