Related papers: Rational functions with only real periodic points

We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary we show that if $X$ is a log…

Given a morphism between smooth projective varieties $f: W \to X$, we study whether $f$-relatively free rational curves imply the existence of $f$-relatively very free rational curves. The answer is shown to be positive when the fibers of…

We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…

Consider a real algebraic curve with set of real points $R\neq\emptyset$ and complexification $P\supset R$. Let $f$ be an algebraic function on $P$ with devisor of critical points $D\subset P$. We prove that $f$ is real after a…

In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real…

We study meromorphic functions in a strip almost periodic with respect to the spherical metric. Then we get a complete description of zeros and poles for this class of functions, find a condition for a meromorphic almost periodic function…

Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f)…

We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…

We study the fibers of a projective morphism and some related algebraic problems. We characterize the analytic spread of a homogeneous ideal through properties of its syzygy matrix. Powers of linearly presented ideals need not be linearly…

In this paper I demonstrate that any pair (m, n) of non-zero and distinct rational numbers may have, at most, four representations as the product of two rational factors such that the sum of factors of m coincides with the sum of factors of…

Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational…

We construct rational elliptic surfaces of index two by explicitly constructing their associated Halphen pencils in the projective plane $\mathbb{P}^2$. For each of the types of singular fibers that occur we construct at least one example…

We show that if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points then it is conjugate to a real meromorphic function after a suitable projective automorphism of the…

For a given elliptic curve, its associated $L$-function evaluated at $1$ is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use…

This is the third of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous two. Let $f:X\to S$ be a map of a smooth projective real algebraic 3-fold to a surface $S$ whose general…

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible…

Let $W$ be a subset of the set of real points of a real algebraic variety $X$. We investigate which functions $f: W \to \mathbb R$ are the restrictions of rational functions on $X$. We introduce two new notions: ${\it curve-rational \,…

We study unramified sections of the fundamental group sequence of smooth projective curves of genus $\geq 2$ over $p$-adic fields together with an integral model. We are particularly interested in the induced specialized sections of the…

We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that…

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…