General classification of rational solutions to A ∘ X = B ∘ Y

Classify all quadruples of rational functions A, B, X, and Y (each of degree at least two over the Riemann sphere) that satisfy the functional equation A ∘ X = B ∘ Y, providing a general description of all rational solutions to this equation.

Background

The paper studies intersections of fields of rational functions via the functional equation A ∘ X = B ∘ Y, which is central to understanding when C(X) ∩ C(Y) is nontrivial and how it is generated.

While many approaches analyze separated-variable curves A(x) − B(y) = 0, they often do not yield direct information about the parametrizing rational functions X and Y. The authors note that beyond the polynomial case (where Ritt theory applies), the general description of rational solutions remains elusive.