Rational Functions on the Projective Line from a Computational Viewpoint

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 $\xi_0, ..., \xi_5$ 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 $\xi_i$, 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.