Classification of rational solutions to B ∘ X = B ∘ Y

Classify all rational functions X and Y (of degree at least two) that solve the equation B ∘ X = B ∘ Y for a given rational function B, thereby determining all triples (B; X, Y) of rational functions satisfying this functional equation.

Background

The authors relate intersections C(X) ∩ C(Y) to the functional equation B ∘ X = B ∘ Y, especially when deg X = 2 and certain invariance conditions hold. In special cases, irreducibility and group actions reduce solutions to compositions with automorphisms.

Despite the importance of understanding when B ∘ X = B ∘ Y has rational or meromorphic solutions (with implications for dynamics and geometry of B), a complete description remains lacking.

References

Since, for a given $B$, the condition that so1 has no solutions in rational or meromorphic functions on $\mathbb{C}$ allows one to solve some non-trivial problems related to the geometry and dynamics of $B$ (see, e.g., , ), the description of rational solutions of so1 is of significant importance. However, the problem remains largely open.

so1:

B∘X=B∘YB \circ X = B \circ Y

— On intersections of fields of rational functions  (2603.29609 - Pakovich, 31 Mar 2026) in Section 3 (Some examples), concluding paragraph