ULethbridge Number Theory and Combinatorics Seminar: Zheng Zhu

Let $f$ be a rational map of degree at least two. The solutions of $fn(x)=α$, as $n$ varies, form an infinite rooted tree on which the absolute Galois group acts. A central question in arithmetic dynamics is when the resulting arboreal Galois representation has finite index in the generic iterated Galois group.

In this talk, I will discuss a connection between this finite-index problem and eventual stability, which asks whether the number of irreducible factors of $fn(x)=α$ remains bounded. Under suitable hypotheses on the critical orbits and geometric monodromy, we establish an equivalence between eventual stability and finite index. The result is conditional on the abc conjecture over number fields and unconditional for non-isotrivial rational maps over characteristic-zero function fields.

The proof combines primitive prime divisors with adelic intersection theory on projective curves. I will also discuss applications to polynomials of arbitrary degree and an explicit family of quartic rational maps whose Galois groups exhibit unusual behavior.

No prior familiarity with arboreal Galois representations will be assumed.