Related papers: Equidistribution and generalized Mahler measures

Let X be a projective integral normal scheme over a number field F; let L be a ample line bundle on X together with a semi-positive adelic metric in the sense of Zhang. The main results of this article are 1) A formula which computes the…

We study the dynamics of stochastic families of rational maps on the projective line. As such families can be infinite and may not typically be defined over a single number field, we introduce the concept of generalized adelic measures,…

In this paper, we prove the equidistribution of periodic points of a regular polynomial automorphism f : A^n -> A^n defined over a number field K: let f be a regular polynomial automorphism defined over a number field K and let v be a prime…

We give a geometric proof that one may compute a particular generalized Mahler integral using equidistribution of preperiodic points of a dynamical system on the sphere. The dynamical system is associated to the multiplication by 2 map on…

Let f be a real or complex polynomial. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of f.

We prove an equidistribution result for the zeros of polynomials with integer coefficients and simple zeros. Specifically, we show that the normalized zero measures associated with a sequence of such polynomials, having small height…

The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial.

The Mahler measure of a polynomial $P$ in $n$ variables is defined as the mean of $\log|P|$ over the $n$-dimensional torus. For certain polynomials with integer coefficients in two variables the Mahler measure is known to be related to…

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…

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree $N$ polynomial chosen…

We establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any value in $\mathbb{C}\setminus\{0\}$ under the derivatives of the iterations of a polynomials $f\in\mathbb{C}[z]$ of degree more than one…

We consider a version of height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler's measure arises in connection with extremal problems for Bergman spaces. It…

Our aim is to explain instances in which the value of the logarithmic Mahler measure of a polynomial can be written in an unexpectedly neat manner. To this end we examine polynomials defining rational curves, which allows their zero-locus…

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…

Following the work of Lal\'in and Mittal on the Mahler measure over arbitrary tori, we investigate the definition of the generalized Mahler measure for all Laurent polynomials in two variables when they do not vanish on the integration…

We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve $L$-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In…

We prove that the local height of a point on a Drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module. This gives rise to a Drinfeld module analog of a weak version of…

In analogy to the equidistribution of preimages of a prescribed point by the iterates of a polynomial map in the complex plane towards the equilibrium measure, we show here the equidistribution of points for which the derivative of the n-th…

A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of an algebraic integer are also discussed.…

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…