Related papers: Generalized R\'edei rational functions and rationa…

Given a rational function of degree at least two defined over a number field k, we study the cardinality of the set of rational iterated preimages. We prove bounds for the cardinality of this set as the rational function varies in certain…

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…

What is ergonomic syntax for relations? In this first paper in a series of two, to answer the question we define regular calculi: a suitably structured functor from a category representing the syntax of regular logic to the category of…

We introduce arrangements of rational sections over curves. They generalize line arrangements on P^2. Each arrangement of d sections defines a single curve in P^{d-2} through the Kapranov's construction of \bar{M}_{0,d+1}. We show a…

We give an explicit description of the arithmetic-geometric extension of iterated Galois groups of rational functions. This yields a complete solution to the extension problem when either the arithmetic or the geometric iterated Galois…

We develop the relation between hyperbolic geometry and arithmetic equidistribution problems that arises from the action of arithmetic groups on real hyperbolic spaces, especially in dimension up to 5. We prove generalisations of Mertens'…

In this paper we present a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. These elliptic units obtained as values of multiple elliptic Gamma functions. These form a…

In this paper, we present a geometric generalization of class field theory, demonstrating how adelic constructions, central to the spectral realization of zeros of L-functions and the geometric framework for explicit formulas in number…

The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…

In the first part of this paper, we develop a general framework that permits a comparison between explicit class field theories for a family of rational function fields $\mathbb{F}_s(t)$ over arbitrary constant fields $\mathbb{F}_s$ and…

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$.…

We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the…

Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential…

Let $\phi$ be an endomorphism of the projective line defined over a global field $K$. We prove a bound for the cardinality of the set of $K$-rational preperiodic points for $\phi$ in terms of the number of places of bad reduction. The…

Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and…

In this technical report we describe a general class of monoids for which (sub)sequential rational can be characterised in terms of a congruence relation in the flavour of Myhill-Nerode relation. The class of monoids that we consider can be…

In this paper, Thue's Fundamentaltheorem is analysed. We show that it includes, and often strengthens, known effective irrationality measures obtained via the so-called hypergeometric method as well as showing that it can be applied to…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…