Rational Functions on the Projective Line from a Computational Viewpoint

TL;DR

The paper advances arithmetic dynamics by combining invariant theory with machine learning to classify automorphism groups of degree $d$ rational maps, focusing on cubic maps as a tractable case. It develops a robust invariant-based framework in the weighted projective moduli space $\mathbb P_{\mathbf{w}}^5(\mathbb Q)$ with $\mathbf{w}=(2,2,3,3,4,6)$, derives explicit invariants $\xi_0,\dots,\xi_5$ and absolute invariants $i_1,\dots,i_5$, and shows that these invariants encode the isomorphism class of a map up to $\mathrm{PGL}_2(k)$-conjugacy. Through a dataset of over two million cubic rational functions with height bound $4$, the authors demonstrate that machine-learning classifiers trained on invariants achieve near-perfect accuracy ($\approx 99.992\%$) for automorphism-group identification, vastly outperforming coefficient-based models on minority classes. The work highlights the power of algebraic invariants to stabilize high-variance classification tasks and lays a path toward extending to higher degrees ($d>3$) and to more complex moduli spaces ($\mathcal M_d^2$), potentially aided by neuro-symbolic methods for invariant computation. Overall, this study provides a scalable, data-augmented algebraic approach to understanding dynamical symmetries of rational maps and their moduli.

Abstract

We explore the moduli space $\mathcal{M}_d^1$ of degree $d \geq 3$ rational functions on the projective line using a machine learning approach, focusing on automorphism group classification. For $d = 3$, where $\mathcal{M}_3^1 = {\mathbb P}_{\mathbf{w}}^5 ({\mathbb Q})$ with weights $\mathbf{w} = (2, 2, 3, 3, 4, 6)$, we generate a dataset of 2,078,697 rational functions over $\Q$ with naive height $\leq 4$. Initial coefficient-based models achieved high overall accuracy but struggled with minority classes due to extreme class imbalance. By using invariants $ξ_0, \ldots, ξ_5$ as features in a Random Forest classifier, we achieved approximately 99.992\% accuracy, mirroring successes in genus 2 curves \cite{2024-03}. This highlights the transformative role of invariants in arithmetic dynamics, yet for $d > 3$, unknown generators of $\mathcal{R}_{(d+1, d-1)}$ pose scalability challenges. Our framework bridges data-driven and algebraic methods, with potential extensions to higher degrees and $\mathcal{M}_d^2$.

Figures (1)

• Figure 1: The inclusions among the loci for degree 3 rational functions

Theorems & Definitions (23)

• Remark 1
• Definition 1
• Lemma 1
• proof
• Definition 2
• Lemma 2
• proof
• Theorem : Clebsch
• Lemma 3
• proof