Rational values of transcendental functions and arithmetic dynamics

TL;DR

This paper investigates algebraic points on transcendental function graphs and uses p-adic methods to establish lower bounds on the degree of splitting fields in polynomial iteration, advancing understanding in arithmetic dynamics.

Contribution

It provides new bounds on algebraic points of transcendental functions and combines complex and p-adic techniques to analyze polynomial iteration in number fields.

Findings

Polynomial bound on algebraic points relative to height and degree

Lower bound on splitting field degree for polynomial iterates

Effective constants depending on polynomial and initial point

Abstract

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with p-adic methods to obtain a lower bound of the form $cD^{n/4 - \varepsilon}$ on the degree of the splitting field of $P^{\circ n}(z)=P^{\circ n}(\alpha)$, where $P$ is a polynomial of degree $D\geq 2$ over a number field, $P^{\circ n}$ is its $n$-th iterate and $c$ depends effectively on $P, \alpha$ and $\varepsilon$. Our $c$ is positive for each algebraic $\alpha$ for which the set $\{P^{\circ n}(\alpha):n\in\mathbb{N}\}$ is infinite.