15 citations · 61 across the 22 of their papers we have counts for
12 papers · 1 filter
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…
Lower bounds on the projective heights of algebraic points
Charles L. Samuels
If are algebraic numbers such that for some integer , then a theorem of Beukers and Zagier gives the best possibl…
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 under certain assumptions on . We prove…
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 for an algebraic number . We show that the computation of $M_\inf…
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…
A collection of metric Mahler measures
Charles L. Samuels
Let denote the Mahler measure of the algebraic number . In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative gro…
The parametrized family of metric Mahler measures
Charles L. Samuels
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 gener…
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 , associated to each positive integer , whose divisibility properties yield a reformulation of the Goldbach conjecture. While this…
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…
The -metric Mahler measures of surds and rational numbers
Charles L. Samuels, Jonas Jankauskas
A. Dubickas and C. Smyth introduced the metric Mahler measure where denotes the u…
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…
Optimal factorizations of rational numbers using factorization trees
Charles L. Samuels, Tanner J. Strunk
Let denote the -metric Mahler measure of the algebraic number . Recent work of the first author established that the infimum in is attained by a single poin…