956 research outputs found
On the heights of totally p-adic numbers
Bombieri and Zannier established lower and upper bounds for the limit infimum
of the Weil height in fields of totally p-adic numbers and generalizations
thereof. In this paper, we use potential theoretic techniques to generalize the
upper bounds from their paper and, under the assumption of integrality, to
improve slightly upon their bounds
Norms extremal with respect to the Mahler measure
In a previous paper, the authors introduced several vector space norms on the
space of algebraic numbers modulo torsion which corresponded to the Mahler
measure on a certain class of numbers and allowed the authors to formulate L^p
Lehmer conjectures which were equivalent to their classical counterparts. In
this paper, we introduce and study several analogous norms which are
constructed in order to satisfy an extremal property with respect to the Mahler
measure. These norms are a natural generalization of the metric Mahler measure
introduced by Dubickas and Smyth. We evaluate these norms on certain classes of
algebraic numbers and prove that the infimum in the construction is achieved in
a certain finite dimensional space.Comment: 24 page
Orthogonal decomposition of the space of algebraic numbers and Lehmer's problem
We introduce vector space norms associated to the Mahler measure by using the
L^p norm versions of the Weil height recently introduced by Allcock and Vaaler.
In order to do this, we determine orthogonal decompositions of the space of
algebraic numbers modulo torsion by Galois field and degree. We formulate L^p
Lehmer conjectures involving lower bounds on these norms and prove that these
new conjectures are equivalent to their classical counterparts, specifically,
the classical Lehmer conjecture in the p = 1 case and the Schinzel-Zassenhaus
conjecture in the p = infinity case.Comment: 29 page
A generalization of Dirichlet's unit theorem
We generalize Dirichlet's -unit theorem from the usual group of -units
of a number field to the infinite rank group of all algebraic numbers
having nontrivial valuations only on places lying over . Specifically, we
demonstrate that the group of algebraic -units modulo torsion is a
\bQ-vector space which, when normed by the Weil height, spans a hyperplane
determined by the product formula, and that the elements of this vector space
which are linearly independent over retain their linear
independence over
Equidistribution and the heights of totally real and totally p-adic numbers
C.J. Smyth was among the first to study the spectrum of the Weil height in
the field of all totally real numbers, establishing both lower and upper bounds
for the limit infimum of the height of all totally real integers and
determining isolated values of the height. Later, Bombieri and Zannier
established similar results for totally p-adic numbers and, inspired by work of
Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use
results on equidistribution of points of low height to generalize both
Bogomolov-type results to a wide variety of heights arising in arithmetic
dynamics
Energy integrals over local fields and global height bounds
We solve an energy minimization problem for local fields. As an application
of these results, we improve on lower bounds set by Bombieri and Zannier for
the limit infimum of the Weil height in fields of totally p-adic numbers and
generalizations thereof. In the case of fields with mixed archimedean and
non-archimedean splitting conditions, we are able to combine our bounds with
similar bounds at the archimedean places for totally real fields.Comment: 13 page
On totally real numbers and equidistribution
C.J. Smyth and later Flammang studied the spectrum of the Weil height in the
field of all totally real numbers, establishing both lower and upper bounds for
the limit infimum of the height of all totally real integers and determining
isolated values of the height. We remove the hypothesis that we consider only
integers and establish an lower bound on the limit infimum of the height for
all totally real numbers. Our proof relies on a quantitative equidistribution
theorem for numbers of small height
Quantitative height bounds under splitting conditions
In an earlier work, the first author and Petsche used potential theoretic
techniques to establish a lower bound for the height of algebraic numbers that
satisfy splitting conditions, such as being totally real or p-adic, improving
on earlier work of Bombieri and Zannier in the totally p-adic case. These
bounds applied as the degree of the algebraic number over the rationals tended
towards infinity. In this paper, we use discrete energy approximation
techniques on the Berkovich projective line to make the dependence on the
degree in these bounds explicit, and we establish lower bounds for algebraic
numbers which depend only on local properties of the numbers.Comment: This second version contains improvements of essentially all
calculations, leading to better bounds in Theorems 1, 7, 10, 11 and 12. Some
errors have been corrected. In particular, the statement from Proposition 14
was not correct in the first version. 22 page
On the non-Archimedean metric Mahler measure
Recently, Dubickas and Smyth constructed and examined the metric Mahler
measure and the metric na\"ive height on the multiplicative group of algebraic
numbers. We give a non-Archimedean version of the metric Mahler measure,
denoted , and prove that if and only if
is a root of unity. We further show that defines a
projective height on as a vector space over . Finally, we
demonstrate how to compute when is a surd
Energy integrals and small points for the Arakelov height
We study small points for the Arakelov height on the projective line. First,
we identify the smallest positive value taken by the Arakelov height, and we
characterize all cases of equality. Next we solve several archimedean energy
minimization problems with respect to the chordal metric on the projective
line, and as an application, we obtain lower bounds on the Arakelov height in
fields of totally real and totally p-adic numbers.Comment: 12 page
- β¦