相关论文: Algebraic values of certain analytic functions def…
We study the set of algebraic numbers of bounded height and bounded degree where an analytic transcendental function takes algebraic values.
We study the set of algebraic numbers of bounded height and bounded degree where an analytic transcendental function takes algebraic values.
We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…
We show that transcendental curves in $\mathbb R^n$ (not necessarily compact) have few rational points of bounded height provided that the curves are well behaved with respect to algebraic sets in a certain sense and can be parametrized by…
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…
We introduce and discuss a new class of (multivalued analytic) transcendental functions which still share with algebraic functions the property that the number of their isolated zeros can be explicitly counted. On the other hand, this class…
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…
For points $(a,b)$ on an algebraic curve over a field $K$ with height $\mathfrak{h}$, the asymptotic relation between $\mathfrak{h}(a)$ and $\mathfrak{h}(b)$ has been extensively studied in diophantine geometry. When $K=\overline{k(t)}$ is…
Transcendental functions, such as exponentials and logarithms, appear in a broad array of computational domains: from simulations in curvilinear coordinates, to interpolation, to machine learning. Unfortunately they are typically expensive…
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…
The classic Schneider-Lang theorem in transcendence theory asserts that there are only finitely many points at which algebraically independent complex meromorphic functions of finite order of growth can simultaneously take values in a…
Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Now, various remarkable transcendental…
It is proved that for all but a finite set of the square-free integers $d$ the value of transcendental function $\exp~(2\pi i ~x+\log\log y)$ is an algebraic number for the algebraic arguments $x$ and $y$ lying in a real quadratic field of…
In this paper, the authors will prove that any subset of $\overline{\QQ}$ can be the exceptional set of some transcendental entire function. Furthermore, we could generalize this theorem to a much more general version and present a unified…
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…
A transcendental function usually returns transcendental values at algebraic points. The (algebraic) exceptions form the so-called \emph{exceptional set}, as for instance the unitary set $\{0\}$ for the function $f(z) = e^z \,$, according…
In the present article we describe a class of algebraic curves on which rational functions of two arguments may reach all their possible limiting values. We also solve a similar question for functions that can be represented as a uniform…
We consider the structure ${\mathbb R}^{\mathrm{RE}}$ obtained from $({\mathbb R},<,+,\cdot)$ by adjoining the restricted exponential and sine functions. We prove Wilkie's conjecture for sets definable in this structure: the number of…
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be…
Holomorphic functions are amazing because their values in an ever so small disk in the complex plane completely determine the function values at arbitrary points in their maximum possible domain. The process of extending such a function…