Independence of multipliers via degenerations of rational maps

Using degenerations of rational maps and local asymptotics of periodic multipliers, we prove that for every $d\ge 2$, the multipliers of any $2d-2$ distinct periodic orbits of degree $d$ rational maps are algebraically independent over $\mathbb C$, provided that at most $d$ of the selected orbits are fixed points. This condition is sharp because of the Holomorphic Index Formula that relates the multipliers of the $d+1$ fixed points. The result of this paper removes the additional restrictions on periods present in earlier work. The proof proceeds by induction on the degree, using a one-hole degeneration for the induction step and a three-hole degeneration in an exceptional degree four case.