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The invariant ring of degree-four rational maps on the projective line

Tanush Shaska

I-AIMS preprint sh-106 · 2026

Let k be an algebraically closed field of characteristic zero. Degree-four rational maps on ℙ¹, up to conjugation, correspond to pairs of binary forms (F, G) ∈ V₅ ⊕ V₃. The associated invariant ring is the joint invariant ring R_(5,3) = k[V₅ ⊕ V₃]^(SL₂). We determine a minimal generating set for R_(5,3), consisting of fifty explicit joint transvectants of degrees at most 18. As consequences we describe the null cone of V₅ ⊕ V₃, construct six explicit absolute invariants that generate the function field of the moduli space ℳ₄¹, and give an effective criterion for determining conjugacy of degree-four rational maps.

MSC classes: Primary 13A50; Secondary 14D22, 14L24, 37P45

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