Upper bounds for the intersection-field index [C(z) : C(X) ∩ C(Y)]

Ascertain whether there exists any general upper bound on the degree [C(z) : C(X) ∩ C(Y)] for rational functions X and Y of degree at least two that satisfy C(X,Y) = C(z) and C(X) ∩ C(Y) ≠ C.

Background

The authors show that even for degree-two rational functions X and Y, the index [C(z) : C(X) ∩ C(Y)] can be arbitrarily large due to dihedral group constructions, indicating that the minimal-degree scenario (equal to deg X * deg Y) does not generally hold.

This raises the broader question of whether any general upper bounds exist for this index under the assumptions C(X,Y) = C(z) and C(X) ∩ C(Y) ≠ C.

References

Although these examples are somewhat specialized, it is unclear whether, for $X$ and $Y$ satisfying us1 and per, there exist any general upper bounds for the degree $[\mathbb{C}(z) : \mathbb{C}(X) \cap \mathbb{C}(Y)].$

us1:

C(X,Y)=C(z),\mathbb{C}(X,Y)=\mathbb{C}(z),

per:

C(X)∩C(Y)≠C  ?\mathbb{C}(X)\cap\mathbb{C}(Y)\neq \mathbb{C}\; ?

— On intersections of fields of rational functions  (2603.29609 - Pakovich, 31 Mar 2026) in Introduction, Section 1 (paragraph discussing degree-two examples and dihedral groups)