Rational Functions on the Projective Line from a Computational Viewpoint

Eslam Badr, Elira Shaska, Tanush Shaska

I-AIMS paper 2024-04 · Published in J. Symbolic Comput. 140 (2027), Paper 102640 · 2027

We develop an explicit invariant-theoretic description of the moduli space ℳ₃¹ of degree-three rational maps on ℙ¹. Using the Clebsch–Gordan decomposition, a cubic map ϕ is represented, up to conjugation, by a pair of binary forms (f, g) ∈ V₄ ⊕ V₂. This identification allows us to construct a system of weighted projective invariants ξ₀, …, ξ₅ of degrees 2, 2, 3, 3, 4, 6, yielding an embedding of ℳ₃¹ into ℙ⁵_((2,2,3,3,4,6)) on the locus where the nonzero coordinates have coprime weights. We further introduce absolute invariants, defined as weight-zero rational functions of the ξᵢ normalized by an invariant I₆, and prove that they uniquely determine the isomorphism class. We determine the algebraic relations among the invariants and obtain a complete stratification of ℳ₃¹ by automorphism groups, allowing the automorphism group of any rational cubic to be recovered directly from its invariants.

MSC classes: 14H10, 13A50, 14L30

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References

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