Andrews–Petsche dynamical Galois-group conjecture

Let $K$ be a global function field of characteristic $p$ , let $f\in K[x]$ have degree $d\ge 2$ , and let $a\in K$ . For $n\ge 1$ , let $K_n$ be the splitting field over $K$ of $f^{\circ n}(x)-a$ , and put $K_\infty=\bigcup_{n\ge 1}K_n$ . Determine precisely when the dynamical Galois group $\operatorname{Gal}(K_\infty/K)$ is abelian. For $d<p$ , the conjectural classification is that this occurs exactly when the pair $(f,a)$ is isotrivial, meaning that there is an affine linear map $\phi\in\overline K[x]$ such that $\phi\circ f\circ\phi^{-1}\in\overline{\mathbb F}_p[x]$ and $\phi(a)\in\overline{\mathbb F}_p$ . In degrees $d\ge p$ , determine the complete classification, including the non-isotrivial examples arising from Drinfeld modules and their associated Lattès maps.

References

Primary source

arXiv

Additional references

Abelian dynamical Galois groups over global function fields — arXiv — Andrea Ferraguti, Patrick Ingram, Carlo Pagano

arithmetic-dynamics function-fields

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new preprint claims the function-field version is proved below the characteristic threshold, but the full classification remains open.

The Andrews–Petsche conjecture predicts that abelian dynamical Galois groups arise only from power-map and Chebyshev-type families, up to the appropriate conjugacy. Andrews and Petsche introduced the conjecture in 2020; subsequent work proves several quadratic, periodic-critical-orbit, and unicritical cases.

Known results

Quadratic polynomials over number fields and global function fields satisfy substantial classifications; the conjecture is complete over $\mathbb{Q}$ in the quadratic case (2020).
Periodic critical orbit cases are proved, as are monic unicritical polynomials over quadratic number fields (Ferraguti–Pagano, 2023).
For each degree $d$ , outside a finite explicit exceptional set of unicritical polynomials, only finitely many basepoints yield abelian groups (Ferraguti–Pagano, 2023).

September 2026 function-field claim

A preprint by Andrea Ferraguti, Patrick Ingram, and Carlo Pagano claims the function-field analogue for polynomial degree $d<p$ , where $p$ is the characteristic, and sharpness at $d=p$ . This is claimed progress toward the conjecture, not an all-degree resolution, and remains unverified.

Current status (as of September 2026): The full all-degree conjecture remains open; the new preprint claims a function-field theorem for $d<p$ with sharpness at $d=p$ , but that claim is unverified.

Sources

No solutions have been posted yet.