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:PF1→PF1.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
