Let K be a number field, let f∈ K(x) have degree at least 2, and let alpha∈P^1(K). Define the backward orbit O^-_f(alpha)=beta∈P^1(overlineK):f^n(beta)=alpha for some n≥ 0. If f is not conjugate over overlineK to a power map, a signed Chebyshev map, or a Lattès map, then O^-_f(alpha)capP^1(K^ab) is finite.

Andrews–Petsche conjecture for rational maps

The conjecture classifies rational dynamical systems over a number field whose backward orbits generate abelian extensions, predicting only the expected power, Chebyshev, and related model maps. Earlier work established substantial special cases but left the general rational-map statement open.

Known results

Andrews–Petsche, 2020: the polynomial conjecture and several cases, including stable quadratic polynomials over $\mathbb{Q}$ .
Ferraguti–Ostafe–Zannier, 2022: finiteness of ramification in abelian backward-orbit extensions and consequences for post-critically finite maps.
2023 work: the conjecture for polynomials with periodic critical orbit and monic unicritical polynomials over quadratic fields.
2024 work: the general polynomial case was still described as open, with additional partial results.

September 2026 claimed resolution

Ferraguti and Pagano’s paper claims the stated number-field and infinite-abelian-backward-orbit forms are settled, via the Bogomolov property of $K^{\mathrm{ab}}$ for canonical heights. The source credits Astra with motivating the key idea, while the mathematical authors are Ferraguti and Pagano.

Current status (as of September 2026): The conjecture is claimed solved in its stated rational-map forms by Ferraguti and Pagano, but independent mathematical verification is not recorded.

Andrews–Petsche conjecture for rational maps

Primary source

Additional references

Progress summary

Solutions 0