Preperiodic integers for $x^d+c$ in large degree
John R. Doyle, Wade Hindes
TL;DR
The paper advances the arithmetic dynamics of the family $f_{d,c}(x)=x^d+c$ in large degree by proving unconditional classifications of preperiodic portraits over number fields, replacing prior abc-conjecture–dependent results with Baker-type bounds on linear forms in logarithms. It shows that for large $d$, there are precisely thirteen preperiodic portraits up to the $d$th roots of unity, provides a finite skeleton of realizable portraits, and derives degree-independent bounds on the number of preperiodic points, with explicit portraits described via the skeleton. It then connects these dynamics results to semigroup dynamics, establishing irreducibility results in semigroups generated by unicritical maps and exploring semigroup PCF-like finite-orbit configurations, including cases with powered fixed points and multiple generators. Overall, the work yields unconditional, quantitative control over preperiodic structure and irreducibility in semigroups, with implications for semigroup dynamics and related Diophantine phenomena.
Abstract
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 thirteen such portraits up to the natural action of roots of unity. In particular, we obtain some of the main results of recent work of the authors unconditionally for algebraic integers by replacing the use of the abc-conjecture with bounds on linear forms in logarithms. We then include applications of this work to several problems in semigroup dynamics, including the construction of irreducible polynomials and the classification of post-critically finite sets.