Related papers: On a problem posed by Mahler
In 1906, Maillet proved that given a non-constant rational function $f$, with rational coefficients, if $\xi$ is a Liouville number, then so is $f(\xi)$. Motivated by this fact, in 1984, Mahler raised the question about the existence of…
In 1902, Paul St\"ackel constructed an analytic function $f(z)$ in a neighborhood of the origin, which was transcendental, and with the property that both $f(z)$ and its inverse, as well as its derivatives, assumed algebraic values at all…
In the early 1900's, Maillet proved that the image of any Liouville number under a rational function with rational coefficients is again a Liouville number. The analogous result for quadratic Liouville matrices in higher dimension turns out…
In this paper, we show the existence of a transcendental function $f\in\mathbb{Z}\{z\}$ with coefficients that are almost all bounded such that $f$ and all its derivatives assume algebraic values at algebraic points. Furthermore, we…
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…
In 1902, P. St\"{a}ckel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at…
In 1902, P. St\"ackel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at…
This note deals with some effective results in Mahler's method. In a recent work, we used a theorem of Philippon to show that given a Mahler function $f(z)$ in ${\bf k}\{z\}$, where ${\bf k}$ denotes a number field, and an algebraic number…
This paper is concerned with Mahler's method. We study in detail the structure of linear relations between values of Mahler functions at algebraic points. In particular, given a field ${\bf k}$, a Mahler function $f(z)\in{\bf k}\{z\}$, and…
In this paper, we shall prove, for any $m\geq 1$, the existence of an uncountable subset of $U$-numbers of type $\leq m$ (which we called the set of {\it $m$-ultra numbers}) for which there exists uncountably many transcendental analytic…
Let $f$ be an $E$-function (in Siegel's sense) not of the form $e^{\beta z}$, $\beta \in \overline{\mathbb{Q}}$, and let $\log$ denote any fixed determination of the complex logarithm. We first prove that there exists a finite set $S(f)$…
We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and…
In this paper, among other things, we prove that any subset of $\overline{\mathbb{Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountable many transcendental entire functions over…
We establish a Liouville-type inequality for the values, at a common nonzero algebraic point, of arbitrary Mahler Mq-functions. As an application, we prove that no such value is a Liouville number, or even a U -number. This solves a…
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…
Suppose that $F(x)\in\mathbb{Z}[[x]]$ is a Mahler function and that $1/b$ is in the radius of convergence of $F(x)$. In this paper, we consider the approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that $F(1/b)$ cannot…
We construct Mahler discrete residues for rational functions and show that they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some…
We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely…
We explore Mahler numbers originating from functions $f(z)$ that satisfy the functional equation $f(z) = (A(z)f(z^d) + C(z))/B(z)$. A procedure to compute the irrationality exponents of such numbers is developed using continued fractions…
In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of…