arXiv subjects
Publications and source records attributed to 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 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.
A. Dubickas and C. Smyth introduced the metric Mahler measure $$ M_1(α) = \inf\left\{\sum_{n=1}^N M(α_n): N \in \mathbb N, α_1 \cdots α_N = α\right\}, $$ where $M(α)$ denotes the usual (logarithmic) Mahler measure of $α\in \overline{\mathbb Q}$. This definition extends in a natural way to the $t$-metric Mahler measure by replacing the sum with the usual $L_t$ norm of the vector $(M(α_1), \dots, M(α_N))$ for any $t\geq 1$. For $α\in \mathbb Q$, we prove that the infimum in $M_t(α)$ 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 $α\in\overline{\mathbb Q}$ by giving an infinite family of quadratic counterexamples. As part of this construction, we provide an explicit formula to compute $M_t(D^{1/k})$ for a square-free $D \in \mathbb N$.
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 point $\barα= (α_1,\ldots,α_N)\in \overline{\mathbb Q}^N$ for all sufficiently large $t$. Nevertheless, no efficient method for locating $\bar α$ is known. In this article, we define a new tree data structure, called a factorization tree, which enables us to find $\barα$ when $α\in \mathbb Q$. We establish several basic properties of factorization trees, and use these properties to locate $\barα$ in previously unknown cases.
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 has the triangle inequality. The construction of $m_t$ is generic, and may be applied to a broader class of functions defined on any Abelian group $G$. We prove analogs of known results with an abstract function on $G$ in place of the Mahler measure. In the process, we resolve an earlier open problem stated by the author regarding $m_t(α)$.
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 $\mathbb P^{N-1}(\overline{\mathbb Q})$.
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 algebraic number $α$ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved as well as establish its analog for the ultrametric Mahler measure.
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 cyclotomic polynomial if and only if there is no representation of $N$ as the sum of two odd primes. The coefficients of these polynomials stabilize, as $N$ grows, to a fixed sequence $a(m)$; they derived upper and lower bounds for $a(m)$, and an asymptotic formula for the summatory function $A(M)$ of the sequence, both under the assumption of a famous conjecture of Hardy and Littlewood. In this article we improve these results: we obtain an asymptotic formula for $a(m)$ under the same assumption, and we establish the asymptotic formula for $A(M)$ unconditionally.