Abstract
Given a field K, a polynomial f∈K[x] of degree d, and a suitable element t∈K, the set of preimages of t under the iterates f∘n carries a natural structure of a d-ary tree. We study conditions under which the absolute Galois group of K acts on the tree by the full group of automorphisms. When d≥20 is even and K=Q we exhibit examples of polynomials with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of K=F(t) and f∈F[x] in which the corresponding Galois groups are the monodromy groups of the ramified covers f∘n:PF 1→PF 1.
Original language | English |
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Pages (from-to) | 416-430 |
Number of pages | 15 |
Journal | Journal of Number Theory |
Volume | 210 |
DOIs | |
State | Published - May 2020 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2019 Elsevier Inc.
Keywords
- Arboreal representation
- Arithmetic dynamics
- Iterated monodromy group