Related papers: On upper bounds of arithmetic degrees

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the…

We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If $X$ is an irreducible affine variety of degree $d\geq 4$ in $\mathbb{A}^n$ which is not the preimage of a curve under a linear map…

We give an upper bound for the number of functionally independent meromorphic first integrals that a discrete dynamical system generated by an analytic map $f$ can have in a neighborhood of one of its fixed points. This bound is obtained in…

Let $E$ be a holomorphic vector bundle on a compact K\"ahler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one defines its…

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

The main purpose of this paper is to define dynamical degrees for rational maps over an algebraic closed field of characteristic zero and prove some basic properties (such as log-concavity) and give some applications. We also define…

If $\alpha_1,\ldots,\alpha_r$ are algebraic numbers such that $$N=\sum_{i=1}^r\alpha_i \ne \sum_{i=1}^r\alpha_i^{-1}$$ for some integer $N$, then a theorem of Beukers and Zagier gives the best possible lower bound on $$\sum_{i=1}^r\log…

Let $X$ be a variety defined over a number field and $f$ be a dominant rational self-map of $X$ of infinite order. We show that $X$ admits many algebraic points which are not preperiodic under $f$. If $f$ were regular and polarized, this…

Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive…

We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an…

In this article we prove several new uniform upper bounds on the number of points of bounded height on varieties over $\mathbb{F}_q[t]$. For projective curves, we prove the analogue of Walsh' result with polynomial dependence on $q$ and the…

In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions, such as being totally real or p-adic, improving on…

We give a mathematical structure on an arithmetic surface, that has algebraic meanings over finite places and can estimate the canonical norm for a relative differential form on the arithmetic surface. This will give a lower bound for the…

Let $d\ge 2$ be an integer, let $c(t)$ be any rational map, and let $f_t(z) := (z^d+t)/z$ be a family of rational maps indexed by t. For each algebraic number $t$, we let $h_{f_t}(c(t))$ be the canonical height of $c(t)$ with respect to the…

We prove that the Weisfeiler-Leman (WL) dimension of the class of all finite planar graphs is at most 3. In particular, every finite planar graph is definable in first-order logic with counting using at most 4 variables. The previously best…

Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence…

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the…

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is…

Let $f$ be an entire transcendental function of finite order and $\Delta$ be a forward invariant bounded Siegel disk for $f$ with rotation number in Herman's class $\mathcal{H}$. We show that if $f$ has two singular values with bounded…

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the…