Related papers: Ax-Schanuel Type Theorems on Functional Transcende…
We classify transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. We focus on the case where the escape is degenerate in the sense that points from…
This survey shows how, for the Nevanlinna class N of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions $ H^\infty$: interpolating…
We present a Phragm\'en-Lindel\"of type theorem with a flavor of Nevanlinna's theorem for subharmonic functions with frequent oscillations between zero and one. We use a technique inspired by a paper of Jones and Makarov.
We discuss the dynamics of semigroups of transcendental entire functions using Fatou-Julia theory and provide a condition for the complete invariance of escaping set and Julia set of transcendental semigroups. Results regarding limit…
The aim of this paper is to prove a theorem of Ax-Lindemann type for complex semi-abelian varieties as an application of a big Picard theorem proved by the author in 1981, and then apply it to prove a theorem of classical Manin-Mumford…
Nevanlinna's five-value theorem is well-known as a famous theorem in value distribution theory, which asserts that two non-constant meromorphic functions on $\mathbb C$ are identical if they share five distinct values ignoring…
For a proper subfield $K$ of $\QQ$ we show the existence of an algebraic number $\alpha$ such that no power $\alpha^n$, $n\geq 1$, lies in $K$. As an application it is shown that these numbers, multiplied by convenient Gaussian numbers, can…
We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the…
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness…
Using tools from the Siegel-Shidlovskii theory of transcendental numbers, we prove that a nontrivial solution of the Airy equation, its derivative, and an antiderivative are algebraically independent over the field of rational functions.…
This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways…
In this paper we give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than $1$ or an entire transcendental function) is connected. The…
In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several…
We study a special class of non-convex functions which appear in nonlinear elasticity; and we prove that they have well-defined Legandre transforms. Several examples are given, and an application to a nonlinear eigenvalue problem
We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasi-projective variety over $\overline{\mathbf{Q}}$ that are integral curves of some algebraic vector field (defined over $\overline{\mathbf{Q}}$).…
For two meromorphic functions $ f $ and $ g $, the equation $ f^m+g^m=1 $ can be regarded as Fermat-type equations. Using Nevanlinna theory for meromorphic functions in several complex variables, the main purpose of this paper is to…
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…
We define the basic pluripotential-theoretic operations in terms of the transcendental theory of non-Archimedean metrics introduced in \cite{DXZ23}. In particular, we establish that the analogue of Boucksom--Jonsson's envelope conjecture…
In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…