Rational Functions on the Projective Line from a Computational Viewpoint
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| Format: | Preprint |
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2025
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| author | Badr, Eslam Shaska, Elira Shaska, Tony |
| author_facet | Badr, Eslam Shaska, Elira Shaska, Tony |
| contents | 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 $ϕ$ 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 $ξ_0, ..., ξ_5$ that embed $\mathcal{M}_3^1$ into $\mathbb{P}^5(2,2,3,3,4,6)$ onto the locus where the gcd of the weights of the non-zero coordinates equals $1$, together with absolute invariants defined as weight-zero rational functions of the $ξ_i$, 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 $ξ_i$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10835 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational Functions on the Projective Line from a Computational Viewpoint Badr, Eslam Shaska, Elira Shaska, Tony Algebraic Geometry 68T07, 68Q32, 37P05, 37P30, 37P45 I.2; I.2.6 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 $ϕ$ 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 $ξ_0, ..., ξ_5$ that embed $\mathcal{M}_3^1$ into $\mathbb{P}^5(2,2,3,3,4,6)$ onto the locus where the gcd of the weights of the non-zero coordinates equals $1$, together with absolute invariants defined as weight-zero rational functions of the $ξ_i$, 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 $ξ_i$. 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. |
| title | Rational Functions on the Projective Line from a Computational Viewpoint |
| topic | Algebraic Geometry 68T07, 68Q32, 37P05, 37P30, 37P45 I.2; I.2.6 |
| url | https://arxiv.org/abs/2503.10835 |