Large arboreal Galois representations

Borys Kadets

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 languageEnglish
Pages (from-to)416-430
Number of pages15
JournalJournal of Number Theory
Volume210
DOIs
StatePublished - May 2020
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

  • Arboreal representation
  • Arithmetic dynamics
  • Iterated monodromy group

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