181 research outputs found
The parametrized family of metric Mahler measures
Let denote the (logarithmic) Mahler measure of the algebraic
number . Dubickas and Smyth, and later Fili and the author, examined
metric versions of . The author generalized these constructions in order to
associate, to each point in , a metric version of the
Mahler measure, each having a triangle inequality of a different strength. We
further examine the functions , using them to present an equivalent form
of Lehmer's conjecture. We show that the function is
constructed piecewise from certain sums of exponential functions. We pose a
conjecture that, if true, enables us to graph for
rational
The Weil height in terms of an auxiliary polynomial
Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give
lower bounds on the Weil height of an algebraic number under certain
assumptions on . We prove a theorem which introduces an auxiliary
polynomial for giving lower bounds on the height of any algebraic number. Our
theorem contains, as corollaries, a slight generalization of the above results
as well as some new lower bounds in other special cases
Estimating heights using auxiliary functions
Several recent papers construct auxiliary polynomials to bound the Weil
height of certain classes of algebraic numbers from below. Following these
techniques, the author gave a general method for introducing auxiliary
polynomials to problems involving the Weil height. The height appears as a
solution to a certain extremal problem involving polynomials. We further
generalize the above techniques to acquire both the projective height and the
height on subspaces in the same way. We further obtain lower bounds on the
heights of points on some subvarieties of
Continued fraction expansions in connection with the metric Mahler measure
The metric Mahler measure was first studied by Dubickas and Smyth in 2001 as
a means of phrasing Lehmer's conjecture in topological language. More recent
work of the author examined a parametrized family of generalized metric Mahler
measures that gives rise to a series of new, and apparently difficult,
problems. We establish a connection between these metric Mahler measures and
the theory of continued fractions in a certain class of special cases. Our
results enable us to calculate metric Mahler measures in several new examples
Metric Mahler measures over number fields
For an algebraic number , the metric Mahler measure was
first studied by Dubickas and Smyth in 2001 and was later generalized to the
-metric Mahler measure by the author in 2010. The definition
of involves taking an infimum over a certain collection
-tuples of points in , and from previous work of
Jankauskas and the author, the infimum in the definition of is
attained by rational points when . As a consequence of our
main theorem in this article, we obtain an analog of this result when is replaced with any imaginary quadratic number field of class number equal
to . Further, we study examples of other number fields to which our methods
may be applied, and we establish various partial results in those cases.Comment: 12 page
The finiteness of computing the ultrametric Mahler measure
Recent work of Fili and the author examines an ultrametric version of the
Mahler measure, denoted for an algebraic number . We
show that the computation of can be reduced to a certain
search through a finite set. Although it is a open problem to record the points
of this set in general, we provide some examples where it is reasonable to
compute and our result can be used to determine
Metric Heights on an Abelian Group
Suppose denotes the Mahler measure of the non-zero algebraic
number . For each positive real number , the author studied a
version of the Mahler measure that has the triangle inequality.
The construction of is generic, and may be applied to a broader class of
functions defined on any Abelian group . We prove analogs of known results
with an abstract function on in place of the Mahler measure. In the
process, we resolve an earlier open problem stated by the author regarding
Lower bounds on the projective heights of algebraic points
If are algebraic numbers such that
for some integer ,
then a theorem of Beukers and Zagier gives the best possible lower bound on
where denotes the Weil Height. We will
extend this result to allow to be any totally real algebraic number. Our
generalization includes a consequence of a theorem of Schinzel which bounds the
height of a totally real algebraic integer
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
The -metric Mahler measures of surds and rational numbers
A. Dubickas and C. Smyth introduced the metric Mahler measure where denotes the usual
(logarithmic) Mahler measure of . This
definition extends in a natural way to the -metric Mahler measure by
replacing the sum with the usual norm of the vector for any . For , we prove that the
infimum in may be attained using only rational points,
establishing an earlier conjecture of the second author. We show that the
natural analogue of this result fails for general by giving an infinite family of quadratic counterexamples. As part of this
construction, we provide an explicit formula to compute for a
square-free
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