Iterates of generic polynomials and generic rational functions
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Abstract:
In 1985, Odoni showed that in characteristic $0$ the Galois group of the $n$-th iterate of the generic polynomial with degree $d$ is as large as possible. That is, he showed that this Galois group is the $n$-th wreath power of the symmetric group $S_d$. We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.References
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Additional Information
- J. Juul
- Affiliation: Department of Mathematics and Statistics, Amherst College, Amherst, Massachusetts 01002
- Email: jamie.l.rahr@gmail.com
- Received by editor(s): May 13, 2015
- Received by editor(s) in revised form: March 16, 2016, and January 22, 2017
- Published electronically: April 25, 2018
- Additional Notes: This work was partially supported by NSF grant DMS-1200749
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 809-831
- MSC (2010): Primary 37P05; Secondary 11G35, 14G25, 12F10
- DOI: https://doi.org/10.1090/tran/7229
- MathSciNet review: 3885162
Dedicated: Dedicated to R.W.K. Odoni