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

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the…

Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and…

We establish a couple of dynamical properties of surjective rational maps $f: X \dashrightarrow X$ for smooth projective surfaces $X$. We also give a numerical characterization of regular $f$ in the case when $X$ is a del Pezzo surface.…

A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally…

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic…

This paper computes the bases of the image of $2$-adic logarithm on the group of the principal units in all 7 quadratic extensions of $\mathbb{Q}_2$. This helps one to understand the free module structure of $2$-adic logarithm at arbitrary…

The $3x+1$ map $T$ is defined on the $2$-adic integers $\mathbb{Z}_2$ by $T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. It is still unproved that under iteration of $T$ the trajectory of any rational $2$-adic integer is eventually…

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 prove the dynamical Mordell-Lang conjecture for product of endomorphisms of an affine curve and a projective curve over $\overline{\mathbb{Q}}$.

This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points. (originally titled ``On Bicritical Rational Maps'' in September 1997; revised and retitled April 1999)

We classify those rational maps $f: \mathbb{P}^1 \to \mathbb{P}^1$ for which there exists a contravariant tensor $q$ which is parallel, i.e. such that $f^*q // q$, by proving that such maps preserve a parabolic orbifold.

We show that any rational cubic hypersurface of dimension at least 33 defined over a number field $K$ vanishes on a $K$-rational projective line, reducing the previous lower bound of Wooley by two. For $K=\mathbb Q$ we can reduce the bound…

We describe a notion of (abstract) projective line over a field as a set equipped with a certain first order structure, and a projectivity between projective lines as a bijection preserving this structure. The structure in question is that…

Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of $q$-ary projective two-weight linear codes is presented, where $q$ is a power of 2. The parameters of…

Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…

This continues the investigation of a combinatorial model for the variation of dynamics in the family of rational maps of degree two, by concentrating on those varieties in which one critical point is periodic. We prove some general results…

In this paper we consider dynamical systems generated by $(3,2)$-rational functions on the field of $p$-adic complex numbers. Each such function has three fixed points. We show that Siegel disks of the dynamical system may either coincide…

We give an example of a dominant rational selfmap of the projective plane whose dynamical degree is a transcendental number.

In the framework of adelic approach we consider real and p-adic properties of dynamical system given by linear fractional map f (x) = (a x + b)/(c x + d), where a, b, c and d are rational numbers. In particular, we investigate behavior of…