📘 Rational functions of the projective line from a computationa viewpoint

📜 Abstract

An explicit invariant-theoretic description of the moduli space \(\mathcal{M}_3^1\) of degree-three rational maps on \(\mathbb{P}^1\) is developed. A cubic map \(\phi\) is represented, up to conjugation, by the pair of binary forms \((f, g) \in V_4 \oplus V_2\) arising from its Clebsch–Gordan decomposition.   From this representation one constructs weighted projective invariants \(\xi_0,\dots,\xi_5\)   embedding \(\mathcal{M}_3^1 \hookrightarrow \mathbb{P}_{(2,2,3,3,4,6)}^5\), together with absolute invariants defined as weight-zero rational functions of the \(\xi_i\), normalized by an additional invariant \(I_6\) of weight 6. These absolute invariants   determine the isomorphism class uniquely.

The stratification of \(\mathcal{M}_3^1\) is described explicitly by equations in the absolute invariants or polynomial relations among the \(\xi_i\). Computational illustrations demonstrate that the resulting invariants provide an effective feature set for automated classification of automorphism groups. The methods suggest natural extensions to higher degrees.

Keywords: Rational functions,  arithmetic dynamics,  machine learning

📝 Bibliographic Metadata

Author: E. Badr, E. Shaska, T. Shaska
Journal: Journal of Symbolic Computation
Year: 2025
DOI: https://doi.org/10.48550/arXiv.2503.10835
Id: 2024-04
Status: in final review

🔗 Links and Resources

PDF: https://www.risat.org/pdf/2024-04.pdf
ArXiv: https://arxiv.org/abs/2503.10835
Data: https://risat.org/software/2024-05     
Code: https://risat.org/software/2024-05     

📋 Citation Information

BibTeX:
@article{[2024-04
 title = {{Rational Functions on the Projective Line from a Computational Viewpoint},
 author = {E. Badr, E. Shaska, and T. Shaska},
 journal = {Journal of Symbolic Computation},
 volume = {Volume Number},
 number = {Issue Number},
 pages = {Page Range},
 year = {YYYY},
 doi = {10.xxxx/xxxx-xxxx},
 eprint = {arXiv:2503.10835},
 eprinttype = {arxiv}
}