Related papers: Preperiodic portraits for unicritical polynomials …

In this follow-up paper, we again inspect a surprising relationship between the set of $m$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient…

Given a number field $K$, we completely classify the preperiodic portraits of the maps $x^d+c$ where $c\in K$ is an algebraic integer and $d$ is sufficiently large depending on the degree of $K$. Specifically, we show that there are exactly…

Let $f$ be a polynomial over a global field $K$. For each $\alpha$ in $K$ and $N$ in $\mathbb{Z}_{\geq 0}$ denote by $K_N(f,\alpha)$ the arboreal field $K(f^{-N}(\alpha))$ and by $D_N(f,\alpha)$ its degree over $K$. It is conjectured that…

Let $K$ be a compact subset of the complex plane $\mathbb C.$ Let $P(K)$ and $R(K)$ be the closures in $C(K)$ of analytic polynomials and rational functions with poles off $K,$ respectively. Let $A(K) \subset C(K)$ be the algebra of…

Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient…

Let $\phi$ be a an endomorphism of degree $d\geq{2}$ of the projective line, defined over a number field $K$. Let $S$ be a finite set of places of $K$, including the archimedean places, such that $\phi$ has good reduction outside of $S$.…

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to…

Fix $d\geq2$ and let $f_{t}(z)=z^{d}+t$ be the family of polynomials parameterized by $t\in\mathbb{C}$. In this article, we will show that there exists a constant $C(d)$ such that for any $a,b\in\mathbb{C}$ with $a^{d}\neq b^{d}$, the…

We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\geq 1$, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a…

Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most $2$. Recently we proved a generalization of…

Let $K$ be the function field of a curve $C$ over a $p$-adic field $k$. We prove that, for each $n, d \geq 1$ and for each hypersurface $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d^2 \leq n$, the second Milnor $K$-theory group of $K$ is…

We establish effective bounds on the number of periodic points of degree-$d$ polynomials $\phi$ defined over $p$-adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials $X^d + c$…

Let $k$ be a field of characteristic different from 2. Let $G$ be a simply connected or adjoint semisimple algebraic $k$-group which does not contain a simple factor of type $E_8$ and such that every exceptional simple factor of type other…

Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q),…

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…

In 2014, Juul, Kurlberg, Madhu and Tucker asked the following: given $K$ a number field and $f$ a rational function with coefficients in $K$, if $f_\mathfrak{p}$ denotes the reduction of $f$ modulo a prime ideal $\mathfrak{p}$ in the ring…

Let K be a number field, let f: P_1 --> P_1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w in P_1(K) are not preperiodic under f. We prove that the set of (m,n) in N^2…

Let $\mathbb K$ be a perfect field of characterstic $p\ge 0$ and let $R\in \mathbb K(x)$ be a rational function. This paper studies the number $\Delta_{\alpha, R}(n)$ of distinct solutions of $R^{(n)}(x)=\alpha$ over the algebraic closure…

To each quadratic number field $K$ and each quadratic polynomial $f$ with $K$-coefficients, one can associate a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, and whose edges reflect the…

For a morphism $f:\P^N \to \P^N$, the points whose forward orbit by $f$ is finite are called preperiodic points for $f$. This article presents an algorithm to effectively determine all the rational preperiodic points for $f$ defined over a…