Rational Functions on the Projective Line from a Computational 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 $\ddot{I}•$ 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 $\hat{I}{¾}_0,...,\hat{I}{¾}_5$ 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 $\hat{I}{¾}_i$, 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 $\hat{I}{¾}_i$. 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.