Related papers: Arithmetic degrees for dynamical systems over func…

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…

Let f be a dominant meromorphic self-map on a compact Kaehler manifold X which preserves a fibration given by a meromorphic map from X to a compact Kaehler manifold Y. We compute the dynamical degrees of f in term of its dynamical degrees…

The dynamics of one dimensional iterative maps in the regime of fully developed chaos is studied in detail. Motivated by the observation of dynamical structures around the unstable fixed point we introduce the geometrical concept of a…

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 article, we consider a dominant rational self-map $f:X \dashrightarrow X$ of a normal projective variety defined over a number field. We study the arithmetic degree $\alpha_k(f)$ for $f$ and $\alpha_k(f,V)$ of a subvariety $V$,…

We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian…

The dynamical structure of the rational map $ax+1/x$ on the projective line over the field Q2 of $2$-adic numbers, is fully described.

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on…

In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on…

We initiate the computability-theoretic study of ringed spaces and schemes. In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures…

We prove that, for every invertible horizontal-like map (i.e., H{\'e}non-like map) in any dimension, the sequence of the dynamical degrees is increasing until that of maximal value, which is the main dynamical degree, and decreasing after…

We consider the class of quasiprojective varieties admitting a dominant morphism onto a curve with negative Euler characteristic. The existence of such a morphism is a property of the fundamental group. We show that for a variety in this…

Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we…

We prove a formula for the multidegrees of a rational map defined by generalized monomials on a projective variety, in terms of integrals over an associated Newton region. This formula leads to an expression of the multidegrees as volumes…

We provide an explicit bound on the number of periodic points of a rational function defined over a number field, where the bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the…

We investigate complete arcs of degree greater than two, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of…

Self-maps everywhere defined on the projective space $\P^N$ over a number field or a function field are the basic objects of study in the arithmetic of dynamical systems. One reason is a theorem of Fakkruddin \cite{Fakhruddin} (with…

We discuss the dynamical, topological, and algebraic classification of rational maps $f$ of the Riemann sphere to itself each of whose critical points $c$ is also a fixed-point of $f$, i.e. $f(c)=c$.

We investigate certain arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic. Parallel to the Plethystic…