Related papers: Mahler's problem and Turyn polynomials
The principal aim of this paper is to establish a purity theorem for Mahler functions that is reminiscent of famous purity theorems for G-functions by D. and G. Chudnovsky and for E-functions (and, more generally, for holonomic arithmetic…
Investigating a problem posed by W. Hengartner (2000), we study the maximal valence (number of preimages of a prescribed point in the complex plane) of logharmonic polynomials, i.e., complex functions that take the form $f(z) = p(z)…
We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there…
The metric Mahler measure was first studied by Dubickas and Smyth in 2001 as a means of phrasing Lehmer's conjecture in topological language. More recent work of the author examined a parametrized family of generalized metric Mahler…
We consider the Mahler measure of the polynomial 1+x_1+x_2+x_3+x_4, which is the first case not yet evaluated explicitly. A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a special value at the point 4 of the…
By using the elementary symmetric polynomials and some results of number theory, we solve the well known problem of Lehmer on Euler's totient function. As application, we obtain a new characterization of prime numbers.
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…
We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let ${\M}_n({\Z})$ denote the monic polynomials of degree $n$ with integer coefficients. A {\it monic integer Chebyshev polynomial} $M_n…
In this work, we are dealing with some properties relating the zeros of a polynomial and its Mahler measure. We provide estimates on the number of real zeros of a polynomial, lower bounds on the distance between the zeros of a polynomial…
We study the Mahler measures of the polynomial family $Q_k(x,y) = x^3+y^3+1-kxy$ using the method previously developed by the authors. An algorithm is implemented to search for CM points with class numbers $\leqslant 3$, we employ these…
In this paper, we consider the relationship between the Mahler measure of a polynomial and its separation. In 1964, Mahler proved that if $f(x) \in \mathbb{Z}[x]$ is separable of degree $n$, then $\operatorname{sep}(f) \gg_n M(f)^{-(n-1)}$.…
Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large…
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.…
We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a…
We study the dynamical Mahler measure of multivariate polynomials and present dynamical analogues of various results from the classical Mahler measure as well as examples of formulas allowing the computation of the dynamical Mahler measure…
We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the growth of homology torsions of finite abelian…
In this paper we prove that the Mahler measures of the Laurent polynomials $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k$, $(x+x^{-1})^2(y+y^{-1})^2(1+z)^3z^{-2}-k$, and $x^4+y^4+z^4+1+k^{1/4}xyz$, for various values of $k$, are of the form $r_1…
After Furstenberg had provided a first glimpse of remarkable rigidity phenomena associated with the joint action of several commuting automorphisms (or endomorphisms) of a compact abelian group, further key examples motivated the…
Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of…
The century old extremal problem, solved by Carath\'eodory and Fej\'er, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the…