Narrow Results

Let $X$ be a complex smooth quasi-projective variety with a fixed epimorphism $\nu :{\pi }_1(X)\twoheadrightarrow \mathbb{Z}$. We consider the asymptotic behavior of invariants such as Betti numbers with any field coefficients and the order of the torsion part of singular integral homology associated to $\nu$, known as the $L^2$-type invariants. At homological degree one, we give concrete formulas for these limits by the geometric information of $X$ when $\nu$ is orbifold effective. The proof relies on a study about cohomological degree one jump loci of $X$. We extend part of Arapura’s result for cohomological degree one jump loci of $X$ with complex field coefficients to the one with positive characteristic field coefficients. As an application, when $X$ is a hyperplane arrangement complement, a combinatoric upper bound is given for the number of parallel positive-dimensional components in cohomological degree one jump loci with complex coefficients. Another application is that we give a positive answer to a question posed by Denham and Suciu for hyperplane arrangement.

We introduce a generalized Mahler measure. It has relations to multiple sine functions and Dirichlet L-functions. In particular, we are able to express special values of Dirichlet L-functions by sum of logarithmic generalized Mahler measures.

Let L = ℓ₁ ∪⋯∪ℓ_d+1 be an oriented link in 𝕊³, and let L(q) be the d-component link ℓ₁ ∪⋯∪ℓ_d regarded in the homology 3-sphere that results from performing 1/q-surgery on ℓ_d+1. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.

The space ${𝒞}$ of conservative vertex colorings (over a field $𝔽$) of a countable, locally finite graph $G$ is introduced. When $G$ is connected, the subspace ${{𝒞}}^0$ of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs $G$ with a cofinite free ${\mathbb{Z}}^d$-action by automorphisms, ${𝒞}$ is dual to a finitely generated module over the polynomial ring $𝔽[x_1^{\pm 1},\dots ,x_d^{\pm 1}]$. Polynomial invariants for this module, the Laplacian polynomials ${\mathrm{\Delta }}_k,k\ge 0$, are defined, and their properties are discussed. The logarithmic Mahler measure of ${\mathrm{\Delta }}_0$ is characterized in terms of the growth of spanning trees.

We present a generalization of the Fox formula for twisted Alexander invariants associated to representations of knot groups over rings of $S$-integers of $F$, where $S$ is a finite set of finite primes of a number field $F$. As an application, we give the asymptotic growth of twisted homology groups.

We obtain explicit lower bounds for the Mahler measure for nonreciprocal polynomials with integer coefficients satisfying certain congruences.

The aim of this note is to prove the Mahler measure identity m(x + x^-1 + y + y^-1 + 5) = 6m(x + x^-1 + y + y^-1 + 1) which was conjectured by Boyd. The proof is achieved by proving relationships between regulators of both curves.

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted M_∞(α) for an algebraic number α. We show that the computation of M_∞(α) can be reduced to a certain search through a finite set. Although it is an 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 M_∞(α).

Given relatively prime polynomials f(x) and g(x) in ℤ[x] with non-zero constant terms, we show that for n greater than an explicitly determined bound depending on f(x) and g(x), if the polynomial f(x)xⁿ + g(x) is non-reciprocal, then its non-cyclotomic part is irreducible except for some explicit cases where a known factorization of f(x)xⁿ + g(x) can easily be described. Prior work of a similar nature is discussed which shows under similar circumstances the non-reciprocal part off(x)xⁿ + g(x) is irreducible. The current paper establishes and makes use of a result which shows that a reciprocal polynomial f(x) with a root off the unit circle must have a root bounded away from the unit circle by an explicitly given function of the degree of f(x), the leading coefficient a of f(x) and the discriminant of f(x). Notably in this result, a need not be 1.

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 for all sufficiently large t. Nevertheless, no efficient method for locating is known. In this paper, we define a new tree data structure, called a factorization tree, which enables us to find when α ∈ ℚ. We establish several basic properties of factorization trees, and use these properties to locate in previously unknown cases.

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their w_n measures.

The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

We study Laurent polynomials in any number of variables that are sums of at most $k$ monomials. We first show that the Mahler measure of such a polynomial is at least $h/2^{k-2}$, where $h$ is the height of the polynomial. Next, restricting to such polynomials having integer coefficients, we show that the set of logarithmic Mahler measures of the elements of this restricted set is a closed subset of the nonnegative real line, with $0$ being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure $1$ is determined by its $k$ coefficients.

We prove an identity between Mahler measures of polynomials that was originally conjectured by Boyd. The combination of this identity with a result of Zudilin leads to a formula involving a Mahler measure of a Weierstrass form of conductor 17 given in terms of $L^{\prime }(E,0)$. Our proof involves a non-trivial identity between regulators which leads to the elliptic curve $L$-function being expressed in terms of the regulator evaluated in a non-rational non-torsion point.

We use the elliptic regulator to recover some identities between Mahler measures involving certain families of genus 2 curves that were conjectured by Boyd and proven by Bertin and Zudilin by differentiating the Mahler measures and using hypergeometric identities. Since our proofs involve the regulator, they yield light into the expected relation of each Mahler measure to special values of $L$-functions of certain elliptic curves.

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients and the number of monomials. In one variable, our result generalizes a classical inequality of Mahler. In $M$ variables, our result depends on ${\mathbb{Z}}^M$ as an ordered group, and in general, our lower bound depends on the choice of ordering.

For each algebraic number $\alpha$ and each positive real number $t$, the $t$-metric Mahler measure $m_t(\alpha )$ creates an extremal problem whose solution varies depending on the value of $t$. The second author studied the points $t$ at which the solution changes, called exceptional points for$\alpha$. Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every $N\in \mathbb{N}$, there exists a number having at least $N$ exceptional points. In this paper, we describe a polynomial time algorithm for establishing the existence of numbers with at least $N$ exceptional points. Our work constitutes an improvement over the best known existing algorithm which requires exponential time. We apply our main result to show that there exist numbers with at least $37$ exceptional points, another improvement over previous work which was only able to reach $11$ exceptional points.

The $k$-generalized Fibonacci and Pell polynomials are the polynomials $X^k-X^{k-1}-X^{k-2}-\cdots -1$ and $X^k-2X^{k-1}-X^{k-2}-\cdots -1$, respectively. Here, $k\ge 2$ is any integer. In this paper, we show that any two roots of some generalized Fibonacci and Pell polynomials are multiplicatively independent confirming a conjecture from [Bravo, Herrera and Luca, Common values of generalized Fibonacci and Pell sequences, J. Number Theory  226 (2021) 51–71].

The minimal integral Mahler measure of a number field $K$, $M({{𝒪}}_K)$, is the minimal Mahler measure of an integral generator of $K$. Upper and lower bounds, which depend on the discriminant and degree of $K$, are known. We show that for three natural families of cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute $M({{𝒪}}_K)$ for all cubics with absolute value of the discriminant bounded by $N$ and show the resulting data for $N=10,000$.

Recently, the second author and Qin numerically verified some Mahler measure identities of genus 2 and 3 polynomial families. In this paper, we use the elliptic regulator to prove an identity involving shifted Mahler measure for Boyd’s family.