📘 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  xxx
 Year: 2025
 DOI: https://doi.org/10.48550/arXiv.2503.10835
 Id: 2024-04
Status: in  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 = {TBA},
  volume = {Volume Number},
  number = {Issue Number},
  pages = {Page Range},
  year = {YYYY},
  doi = {10.xxxx/xxxx-xxxx},
  eprint = {arXiv:2503.10835},
  eprinttype = {arxiv}
 }

📋 History of the paper and reviews