Related papers: M\'ethode de Mahler: relations lin\'eaires, transc…

Mahler equations arise in a wide range of contexts including the study of finite automata, regular sequences, algebraic series over Fp(z), and periods of Drinfeld modules. Introduced a century ago by K. Mahler to study the transcendence of…

Let $M(\alpha)$ denote the (logarithmic) Mahler measure of the algebraic number $\alpha$. Dubickas and Smyth, and later Fili and the author, examined metric versions of $M$. The author generalized these constructions in order to associate,…

In this note, the main focus is on a question about transcendental entire functions mapping $\mathbb{Q}$ into $\mathbb{Q}$ (which is related to a Mahler's problem). In particular, we prove that, for any $t>0$, there is no a transcendental…

We study real numbers defined by multidimensional automatic arrays weighted by multiplicatively independent bases. Let $a_1, \dots, a_r\geq 2$ be integers such that $\log a_1, \dots, \log a_r$ are $\mathbb Q$-linearly independent. Given…

We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behaviour of a Mahler function $f(z)$ as $z$ goes radially to a root of unity to deduce algebraic…

Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the…

Let $\alpha$ and $\beta$ be positive real numbers. Let $F(x) \in K[[x^\Gamma]]$ be a Hahn series. We prove that if $F(x)$ is both $\alpha$-Mahler and $\beta$-Mahler then it must be a rational function, $F(x) \in K(x)$, assuming some…

For an algebraic number $\alpha$ we denote by $M(\alpha)$ the Mahler measure of $\alpha$. As $M(\alpha)$ is again an algebraic number (indeed, an algebraic integer), $M(\cdot)$ is a self-map on $\overline{\mathbb{Q}}$, and therefore defines…

We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of…

A 1-period is a complex number given by the integral of a univariate algebraic function, where all data involved -- the integrand and the domain of integration -- are defined over algebraic numbers. We give an algorithm that, given a finite…

We develop a theory of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence, which also includes the possibility of dealing with several systems associated with sufficiently independent…

For an algebraic number $\alpha$, the metric Mahler measure $m_1(\alpha)$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$-metric Mahler measure $m_t(\alpha)$ by the author in 2010. The definition of…

Let $K$ be a field of characteristic 0, $f:\mathbb{N}\to K$ be a multiplicative function, and $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ be algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function…

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted $M_\infty(\alpha)$ for an algebraic number $\alpha$. We show that the computation of $M_\infty(\alpha)$ can be reduced to a certain search…

After H\"older proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic…

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…

In the frame of Mahler's method for algebraic independence we show that the algebraic relations over Q linking the values of functions solutions of a system of functional equations come from the algebraic relations between the functions…

In this paper, we shall prove that any subset of $\overline{\mathbb Q}\cap B(0,1)$, which is closed under complex conjugation and which contains the element $0$, is the exceptional set of uncountably many transcendental functions, analytic…

The aim of this paper is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of differential fields and properties of…

We show that $T_p(z)=\prod_{j=1}^{\infty}(1-z^{p^{j}})^{-1/p^{j}}$ is transcendental over $\overline{\mathbb{Q}}(z)$, and establish the transcendence of its values at nonzero algebraic points inside the unit disk. Furthermore, we obtain an…