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We provide a general result for the algebraic independence of Mahler functions by a new method based on asymptotic analysis. As a consequence of our method, these results hold not only over $\mathbb{C}(z)$, but also over…

We show that there exists an entire function f without zeros for which the associated Newton function N(z)=z-f(z)/f'(z) is a transcendental meromorphic functions without Baker domains. We also show that there exists an entire function f…

In this article, which is dedicated to my friend and colleague Boris Zilber on the occasion of his 75th birthday, I put forward a strategy for proving his quasiminimality conjecture for the complex exponential field. That is, for showing…

In order to analyse the way in which the size of the iterates $(f^n(z))$ of a transcendental entire function $f$ can behave, we introduce the concept of the {\it annular itinerary} of a point $z$. This is the sequence of non-negative…

Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2…

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…

In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative $S(f)$ of a locally univalent analytic function $f$ in the unit disk satisfies that $\limsup_{|z|\to 1} |S(f)(z)| (1-|z|^2)^2 < 2$, then there exists a positive…

During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares (posed by Pietro Mengoli in mid 17th century). In the words of Andre Weil (1989) "as…

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…

About fifty years ago Mahler proved that if $\alpha>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $\alpha^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $\alpha$ and $l$. Answering…

We study the rationality of the Artin-Mazur zeta function of a dynamical system defined by a polynomial self-map of A^1(k), where k is the algebraic closure of the finite field F_p. The zeta functions of the maps f(x)=x^m for (p,m)=1 and…

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…

The main purpose of this article is to provide new results on algebraic independence of values of Mahler functions and their generalizations. Simultaneously, we establish new measures of algebraic independence for these values. Among the…

E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with polynomial coefficients. They were introduced by Siegel in 1929…

In this paper we introduce a new infinite set of transcendental integrals. Each of them is expressed by corresponding value of the function $|\zf|^{-2}$. Such a property is another argument about universality of the Riemann zeta-function…

Marx and Strohh\"acker showed around in 1933 that $f(z)/z$ is subordinate to $1/(1-z)$ for a normalized convex function $f$ on the unit disk $|z|<1.$ Brickman, Hallenbeck, MacGregor and Wilken proved in 1973 further that $f(z)/z$ is…

Let $t\geq2$ and $k\geq1$ be integers. Let $H_{k}(z)$ with $\left\vert z\right\vert <1$ be the limit of a certain subsequence of the Stern polynomials introduced by Dilcher and Eriksen. We use Mahler's method to prove the algebraic…

Let ${\mathcal A}$ be the class of functions analytic in the unit disk ${\mathbb D} := \{ z\in {\mathbb C}:\, |z| < 1 \}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we study the class $\mathcal{U}(\lambda)$,…

We discuss the possibility of constructing a function that validates the definition or not definition of the partial recursive functions of one variable. This is a topic in computability theory, which was first approached by Alan M. Turing…

Let K be the scalar field of the first orthomodular (or Form Hilbert) space, described by H. Keller in 1980. It has a non-Archimedean order, an infinite rank valuation compatible with the order as well as an explicitly defined ultrametric,…