Related papers: The Mahler Measure of Parametrizable Polynomials

This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by $P_d, d\geq 1$, unbounded in both degree and genus. By using a closed formula for the Mahler measure introduced in "Volume function and Mahler…

We exhibit some nontrivial evaluations of the areal Mahler measure of multivariable polynomials, defined by Pritsker [Pri08] by considering the integral over the product of unit disks instead of the unit torus as in the standard case. As in…

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…

Let g be a nonconstant rational map from the projective line to itself that has degree greater than one and is defined over a number field. The map g gives rise to generalized Mahler measures for polynomials in one variable. We use…

We exhibit a change of variables that maintains the Mahler measure of a given polynomial. This method leads to the construction of highly non-trivial polynomials with given Mahler measure and settles some conjectural numerical formulas due…

We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous…

We solve a Lehmer-type question about the Mahler measure of integer-valued polynomials.

The Mahler measure of the polynomials $t(x^m-1) y - (x^n-1) \in \dC[x,y]$ is essentially the sum of volumes of a certain collection of ideal hyperbolic polyhedra in $\HH^3$, which can be determined a priori as a function on the parameter…

There are many examples of several-variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger, and…

Motivated by applications of algebraic geometry, we introduce the Galois width, a quantity characterizing the complexity of solving algebraic equations in a restricted model of computation allowing only field arithmetic and adjoining…

We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of…

We introduce the ratio of the number of roots of a polynomial $P_{d}$, less than one in modulus, to its degree $d$ as an alternative to Mahler measure. We investigate some properties of the alternative. We generalise this definition for a…

In this work we apply the techniques that were developed in [Lalin: An algebraic integration for Mahler measure] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of…

In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of…

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show…

If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version…

We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and…

Given fields $k \subseteq L$, our results concern one parameter $L$-parametric polynomials over $k$, and their relation to generic polynomials. The former are polynomials $P(T,Y) \in k[T][Y]$ of group $G$ which parametrize all Galois…

Mahler's problem asks for the largest possible value of the Mahler measure, normalized by the $L_2$ norm, of a polynomial with $\pm1$ coefficients and large degree. We establish a new record value in this problem exceeding $0.95$ by…

The system of $N$ classical particles on the line with the Weierstrass $\wp$ function as potential is known to be completely integrable. Recently D'Hoker and Phong found a beautiful parameterization by the polynomial of degree $N$ of the…