Related papers: Rational map $ax+1/x$ on the projective line over …

We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$…

For any prime $p\ge 5$, we show that generic hypersurface $X_p\subset\mathbb{P}^p$ defined over $\mathbb{Q}$ admits a non-trivial rational dominant self-map of degree $>1$, defined over $\bar{\mathbb{Q}}$. A simple arithmetic application of…

Sometimes we obtain attractive results when associating facts to simple elements. The goal of this work is to introduce a possible alternative in the study of the dynamics of rational maps.

A twisted rational map over a non-archimedean field $K$ is the composition of a rational function over $K$ and a continuous automorphism of $K$. We explore the dynamics of some twisted rational maps on the Berkovich projective line.

We show that the proportion of plane cubic curves over ${\mathbb Q}_p$ that have a ${\mathbb Q}_p$-rational point is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function…

We study arithmetic degree of a dominant rational self-map on a smooth projective variety over a function field of characteristic zero. We see that the notion of arithmetic degree and some related problems over function fields are…

We classify the possible torsion structures of rational elliptic curves over sextic number fields.

In this paper, we show that projective globally $F$-regular threefolds, defined over an algebraically closed field of characteristic $p\geq 11$, are rationally chain connected.

Let G/Q be an homogeneous variety embedded in a projective space P thanks to an ample line bundle L. Take a projective space containing P and form the cone X over G/Q, we call this a cone over an homogeneous variety. Let $\alpha$ a class of…

We produce an infinite family of transcendental numbers which, when raised to their own power, become rational. We extend the method, to investigate positive rational solutions to the equation $x^x = \alpha$, where $\alpha$ is a fixed…

For a rational function of several variables with nonnegative imaginary part on the upper poly-half-plane, the matrix representations are obtained.

We study the set $R$ of nonplanar rational curves of degree $d<q+2$ on a smooth Hermitian surface $X$ of degree $q+1$ defined over an algebraically closed field of characteristic $p>0$, where $q$ is a power of $p$. We prove that $R$ is the…

We introduce a class of rational functions $A:\,\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ which can be considered as a natural extension of the class of Latt\`es maps and establish basic properties of functions from this class.

Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a…

We compute the set of naive pointed homotopy classes of endomorphisms of the projective line P^1 over the spectrum of a field. Our computation compares well with Fabien Morel's one of the motivic pointed homotopy classes of endomorphisms of…

We construct a family of rational map sequences providing an arbitrary large number of dynamically independent rescaling limits of non monomial type. From this, we deduce the existence of a family of rational maps providing a non trivial…

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification,…

For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.

The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. Unnoticed since then, natural questions related to representations in that specific…

A telegraphic survey of some of the standard results and conjectures about the set $C({\bf Q})$ of rational points on a smooth projective absolutely connected curve $C$ over ${\bf Q}$.