C. Samuels

activity

20102023

most citedOn the non-Archimedean metric Mahler measure

15 citations · 61 across the 22 of their papers we have counts for

collaborators

Showing math.NTShow all

22 papers · 1 filter

math.NT2023

Consistent maps and their associated dual representation theorems

Charles L. Samuels

A 2009 article of Allcock and Vaaler examined the vector space $G := Q^{\times} / Q_{tors}^{\times}$ over $Q$ , describing…

math.NT2021

A classification of $Q$ -valued linear functionals on $Q^{\times}$ modulo units

Charles L. Samuels

Let $Q$ be an algebraic closure of $Q$ and let $A$ denote the ring of algebraic integers in $Q$ . If $\mathcal S = \overline{\mathbb Q}…

math.NT2021

A Polynomial Time Test to Detect Number with Many Exceptional Points

Ryan Carpenter, Charles L. Samuels

For each algebraic number $α$ and each positive real number $t$ , the $t$ -metric Mahler measure $m_{t} (α)$ creates an extremal problem whose solution varies depending on the value of…

math.NT2018

Rotation Symmetries of Sequential Matrices with Applications to the Jacobi Symbol

Yemeen Ayub, Charles L. Samuels

Suppose that $p$ is an odd prime and $(\frac{\cdot}{p})$ denotes the Legendre symbol modulo $p$ . If $p$ is has the form $p = n^{2} + 1$ then one easily verifies that $\genfra…

math.NT2017★ 3 cited

Direct Limits of Adèle Rings and Their Completions

James P. Kelly, Charles L. Samuels

The adèle ring $A_{K}$ of a global field $K$ is a locally compact, metrizable topological ring which is complete with respect to any invariant metric on $A_{K}$ . For a…

math.NT2017

Metric Mahler measures over number fields

Charles L. Samuels

For an algebraic number $α$ , the metric Mahler measure $m_{1} (α)$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$ -metric Mahler measure $m_{t} (α)$ …

math.NT2016★ 2 cited

Counting Exceptional Points for Rational Numbers Associated to the Fibonacci Sequence

Charles L. Samuels

If $α$ is a non-zero algebraic number, we let $m (α)$ denote the Mahler measure of the minimal polynomial of $α$ over $Z$ . A series of articles by Dubickas and Smyth, and la…

math.NT2015★ 3 cited

Continued fraction expansions in connection with the metric Mahler measure

Charles L. Samuels

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 examine…

math.NT2014★ 15 cited

On the non-Archimedean metric Mahler measure

Paul Fili, Charles L. Samuels

Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a non-Archimed…

math.NT2014★ 5 cited

Lower bounds on the projective heights of algebraic points

Charles L. Samuels

If $α_{1}, \dots, α_{r}$ are algebraic numbers such that $N = i = 1 \sum r α_{i} \neq = i = 1 \sum r α_{i}^{- 1}$ for some integer $N$ , then a theorem of Beukers and Zagier gives the best possibl…

math.NT2014

The Weil height in terms of an auxiliary polynomial

Charles L. Samuels

Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number $α$ under certain assumptions on $α$ . We prove…

math.NT2014★ 1 cited

The finiteness of computing the ultrametric Mahler measure

Charles L. Samuels

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted $M_{\infty} (α)$ for an algebraic number $α$ . We show that the computation of $M_\inf…

math.NT2014★ 1 cited

Two inequalities on the areal Mahler measure

Stephen K. K. Choi, Charles L. Samuels

Recent work of Pritsker defines and studies an areal version of the Mahler measure. We further explore this function with a particular focus on the case where its value is small, a…

math.NT2014

A collection of metric Mahler measures

Charles L. Samuels

Let $M (α)$ denote the Mahler measure of the algebraic number $α$ . In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative gro…

math.NT2014★ 10 cited

The parametrized family of metric Mahler measures

Charles L. Samuels

Let $M (α)$ 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 gener…

math.NT2014

Polynomials whose reducibility is related to the Goldbach conjecture

Peter B. Borwein, Stephen K. K. Choi, Greg Martin +1

We introduce a collection of polynomials $F_{N}$ , associated to each positive integer $N$ , whose divisibility properties yield a reformulation of the Goldbach conjecture. While this…

math.NT2014★ 2 cited

Estimating heights using auxiliary functions

Charles L. Samuels

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 gen…

math.NT2014★ 7 cited

The $t$ -metric Mahler measures of surds and rational numbers

Charles L. Samuels, Jonas Jankauskas

A. Dubickas and C. Smyth introduced the metric Mahler measure $M_{1} (α) = in f {n = 1 \sum N M (α_{n}) : N \in N, α_{1} \dots α_{N} = α},$ where $M (α)$ denotes the u…

math.NT2014★ 5 cited

The infimum in the metric Mahler measure

Charles L. Samuels

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…

math.NT2014★ 1 cited

Optimal factorizations of rational numbers using factorization trees

Charles L. Samuels, Tanner J. Strunk

Let $m_{t} (α)$ denote the $t$ -metric Mahler measure of the algebraic number $α$ . Recent work of the first author established that the infimum in $m_{t} (α)$ is attained by a single poin…

math.NT2012★ 6 cited

Metric Heights on an Abelian Group

Charles L. Samuels

Suppose $m (α)$ denotes the Mahler measure of the non-zero algebraic number $α$ . For each positive real number $t$ , the author studied a version $m_{t} (α)$ of the Mahler measure that…

math.NT2010

The size of coefficients of certain polynomials related to the Goldbach conjecture

Greg Martin, Charles L. Samuels

Recent work of Borwein, Choi, and the second author examined a collection of polynomials closely related to the Goldbach conjecture: the polynomial $F_{N}$ is divisible by the $N$ th…

Consistent maps and their associated dual representation theorems

A classification of Q-valued linear functionals on Q​× modulo units

A Polynomial Time Test to Detect Number with Many Exceptional Points

Rotation Symmetries of Sequential Matrices with Applications to the Jacobi Symbol

Direct Limits of Adèle Rings and Their Completions

Metric Mahler measures over number fields

Counting Exceptional Points for Rational Numbers Associated to the Fibonacci Sequence

Continued fraction expansions in connection with the metric Mahler measure

On the non-Archimedean metric Mahler measure

Lower bounds on the projective heights of algebraic points

The Weil height in terms of an auxiliary polynomial

The finiteness of computing the ultrametric Mahler measure

Two inequalities on the areal Mahler measure

A collection of metric Mahler measures

The parametrized family of metric Mahler measures

Polynomials whose reducibility is related to the Goldbach conjecture

Estimating heights using auxiliary functions

The t-metric Mahler measures of surds and rational numbers

The infimum in the metric Mahler measure

Optimal factorizations of rational numbers using factorization trees

Metric Heights on an Abelian Group

The size of coefficients of certain polynomials related to the Goldbach conjecture

A classification of $Q$ -valued linear functionals on $Q^{\times}$ modulo units

The $t$ -metric Mahler measures of surds and rational numbers