A Genetic Algorithm for Generating Extreme Examples in Arithmetic Dynamics
Abstract
We describe a genetic algorithm to find extreme examples in the arithmetic of dynamical systems. The algorithm is applied to four problems: small (non-zero) canonical heights, many rational preperiodic points, long rational cycles, and long rational tails. Data is provided for extreme examples generated for polynomials up to degree 13 and rational functions up to degree 5. This work significantly expands the known examples of extreme behavior for several of the conjectured behaviors in arithmetic dynamics and provides a foundation from which to begin a more advanced application of machine learning techniques in the creation of extreme examples for arithmetic dynamics.
Cite
@article{arxiv.2601.11482,
title = {A Genetic Algorithm for Generating Extreme Examples in Arithmetic Dynamics},
author = {Benjamin Hutz},
journal= {arXiv preprint arXiv:2601.11482},
year = {2026}
}
Comments
updated version. Fixed several small errors in the data. article 15 pages. Appendix with data tables 43 pages. github for code: https://github.com/bhutz/GA_for_ADS_examples