Preperiodic integers for x^d+c in large degree

2025/10/15 by Doyle, John R., Hindes, Wade
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.14067

Abstract

Given a number field K, we completely classify the preperiodic portraits of the maps x^d+c where c∈ 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.

Citations

• Unicritical polynomials over abc-fields: from uniform boundedness to dynamical Galois groups
• On the proportion of irreducible polynomials in unicritically generated semigroups
• Irreducible polynomials in quadratic semigroups
• Galois groups and prime divisors in random quadratic sequences
• The Uniform Boundedness and Dynamical Lang Conjectures for polynomials
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• Orbit counting in polarized dynamical systems
• Dynamical uniform boundedness and the abc-conjecture
• The abc-Conjecture implies uniform bounds on dynamical Zsigmondy sets
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• Finite ramification for preimage fields of postcritically finite morphisms
• Preperiodic points for rational functions defined over a global field in\n terms of good reductions
• A finiteness result for post-critically finite polynomials
• Lower bounds on the canonical height associated to the morphism ϕ(z)= z^d+c
• Preperiodic points of polynomials over global fields
• Finitely ramified iterated extensions

Related

• Unicritical polynomials over abc-fields: from uniform boundedness to dynamical Galois groups
• On algebraic integers of bounded house and preperiodicity in polynomial\n semigroup dynamics
• Preperiodic points of polynomials over global fields
• On the abc Conjecture in Algebraic Number Fields
• On the number of quadratic polynomials with a given portrait