2025/03/13 by Eslam Badr, Badr, Eslam, Elira Shaska +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #37P05 #37P30 #37P45 #68Q32 #68T07 #Algebraic Geometry (math.AG) #FOS: Mathematics #I.2 #I.2.6 #Model Reduction and Neural Networks #Polynomial and algebraic computation #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2503.10835
openalex publication_date 2025/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An explicit invariant-theoretic description of the moduli space M31 of degree-three rational maps on ℙ1 is developed. A cubic map ϕ is represented, up to conjugation, by the pair of binary forms (f, g) ∈ V4 ⊕ V2 arising from its Clebsch--Gordan decomposition. From this representation one constructs weighted projective invariants ξ0, ..., ξ5 that embed M31 into ℙ5(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 ξi, normalized by an additional invariant I6 of weight 6. These absolute invariants determine the isomorphism class uniquely. The stratification of M31 is described explicitly by equations in the absolute invariants or polynomial relations among the ξi. 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.