Let $\varphi :X\to X$ be a morphism of a variety defined over a number field $K$, let $V\subset X$ be a $K$-subvariety, and let ${𝒪}_{\varphi }\left(P\right)=\{{\varphi }^n\left(P\right):n\ge 0\}$ be the orbit of a point $P\in X\left(K\right)$. We describe a local-global principle for the intersection $V\cap {𝒪}_{\varphi }\left(P\right)$. This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of $V\left(K\right)$ are Brauer–Manin unobstructed for power maps on ${\mathbb{P}}^2$ in two cases: (1) $V$ is a translate of a torus. (2) $V$ is a line and $P$ has a preperiodic coordinate. A key tool in the proofs is the classical...

We consider two issues concerning polynomial cycles. Namely, for a discrete valuation domain $R$ of positive characteristic (for $N\ge 1$) or for any Dedekind domain $R$ of positive characteristic (but only for $N\ge 2$), we give a closed formula for a set $𝒞YCL(R,N)$ of all possible cycle-lengths for polynomial mappings in $R^N$. Then we give a new property of sets $𝒞YCL(R,1)$, which refutes a kind of conjecture posed by W. Narkiewicz.

Bombieri and Zannier established lower and upper bounds for the limit infimum of the Weil height in fields of totally $p$-adic numbers and generalizations thereof. In this paper, we use potential theoretic techniques to generalize the upper bounds from their paper and, under the assumption of integrality, to improve slightly upon their bounds.

In this paper, the local convergence analysis of the family of Kung-Traub's two-point method and the convergence ball for this family are obtained and the dynamical behavior on quadratic and cubic polynomials of the resulting family is studied. We use complex dynamic tools to analyze their stability and show that the region of stable members of this family is vast. Numerical examples are also presented in this study. This method is compared with several widely used solution methods by solving test...

Let $f$ be a polynomial of degree at least 2 with coefficients in a number field $K$, let $x_0$ be a sufficiently general element of $K$, and let $\alpha$ be a root of $f$. We give precise conditions under which Newton iteration, started at the point $x_0$, converges $v$-adically to the root $\alpha$ for infinitely many places $v$ of $K$. As a corollary we show that if $f$ is irreducible over $K$ of degree at least 3, then Newton iteration converges $v$-adically to any given root of $f$ for infinitely many places $v$. We also conjecture that...