On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps

TL;DR

This paper provides computational evidence supporting Poonen's conjecture that quadratic polynomials over rationals have at most 9 rational periodic points, verifying the conjecture for many parameters and exploring bounds over quadratic fields.

Contribution

The paper offers extensive computational verification of Poonen's conjecture for c values up to 10^8 and investigates bounds on periods over quadratic number fields.

Findings

Verified Poonen's conjecture for c up to height 10^8

Suggested the maximum period over quadratic fields is 6

Provided evidence supporting the conjecture's validity

Abstract

The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. In particular, Poonen conjectured that there are at most 9 periodic points defined over the rational numbers for any map in the family x^2 + c for c rational. We verify this conjecture for c values up to height 10^8. For quadratic number fields, we provide evidence that the upper bound on the exact period of Q-rational periodic point is 6.