Iterates of generic polynomials and generic rational functions

Juul, J.

doi:10.1090/tran/7229

Iterates of generic polynomials and generic rational functions

Author: J. Juul
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
Published electronically: April 25, 2018
MathSciNet review: 3885162
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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 This work was partially completed by the late R.W.K. Odoni in an unpublished paper.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

Nigel Boston and Rafe Jones, The image of an arboreal Galois representation, Pure Appl. Math. Q. 5 (2009), no. 1, 213–225. MR 2520459, DOI https://doi.org/10.4310/PAMQ.2009.v5.n1.a6
Stephen D. Cohen, The distribution of polynomials over finite fields. II, Acta Arith. 20 (1972), 53–62. MR 291135, DOI https://doi.org/10.4064/aa-20-1-53-62
Stephen D. Cohen, Permutation polynomials and primitive permutation groups, Arch. Math. (Basel) 57 (1991), no. 5, 417–423. MR 1129514, DOI https://doi.org/10.1007/BF01246737
S. D. Cohen and R. W. K. Odoni, The Farey density of norm subgroups of global fields. II, Glasgow Math. J. 18 (1977), no. 1, 57–67. MR 432597, DOI https://doi.org/10.1017/S0017089500003037
John Cullinan and Farshid Hajir, Ramification in iterated towers for rational functions, Manuscripta Math. 137 (2012), no. 3-4, 273–286. MR 2875279, DOI https://doi.org/10.1007/s00229-011-0460-y
Michael Fried, On a conjecture of Schur, Michigan Math. J. 17 (1970), 41–55. MR 257033
Michael D. Fried and Moshe Jarden, Field arithmetic, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2008. Revised by Jarden. MR 2445111
Arnaldo Garcia and Henning Stichtenoth (eds.), Topics in geometry, coding theory and cryptography, Algebra and Applications, vol. 6, Springer, Dordrecht, 2007. MR 2265387
Robert M. Guralnick, Thomas J. Tucker, and Michael E. Zieve, Exceptional covers and bijections on rational points, Int. Math. Res. Not. IMRN 1 (2007), Art. ID rnm004, 20. MR 2331902, DOI https://doi.org/10.1093/imrn/rnm004
Spencer Hamblen, Rafe Jones, and Kalyani Madhu, The density of primes in orbits of $z^d+c$, Int. Math. Res. Not. IMRN 7 (2015), 1924–1958. MR 3335237, DOI https://doi.org/10.1093/imrn/rnt349
Gerald J. Janusz, Algebraic number fields, 2nd ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, Providence, RI, 1996. MR 1362545
Rafe Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 523–544. MR 2439638, DOI https://doi.org/10.1112/jlms/jdn034
Rafe Jones and Michelle Manes, Galois theory of quadratic rational functions, Comment. Math. Helv. 89 (2014), no. 1, 173–213. MR 3177912, DOI https://doi.org/10.4171/CMH/316
Jamie Juul, Pär Kurlberg, Kalyani Madhu, and Tom J. Tucker, Wreath products and proportions of periodic points, Int. Math. Res. Not. IMRN 13 (2016), 3944–3969. MR 3544625, DOI https://doi.org/10.1093/imrn/rnv273
Vijaya Kumar Murty and John Scherk, Effective versions of the Chebotarev density theorem for function fields, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 6, 523–528 (English, with English and French summaries). MR 1298275
Serge Lang, Algebraic numbers, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto-London, 1964. MR 0160763
Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, American Mathematical Society, Providence, RI, 2005. MR 2162164
R. W. K. Odoni, The Galois theory of iterates and composites of polynomials, Proc. London Math. Soc. (3) 51 (1985), no. 3, 385–414. MR 805714, DOI https://doi.org/10.1112/plms/s3-51.3.385
G. Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), no. 1, 145–254 (German). MR 1577579, DOI https://doi.org/10.1007/BF02546665
P. Stevenhagen and H. W. Lenstra Jr., Chebotarëv and his density theorem, Math. Intelligencer 18 (1996), no. 2, 26–37. MR 1395088, DOI https://doi.org/10.1007/BF03027290
Henning Stichtenoth, Algebraic function fields and codes, 2nd ed., Graduate Texts in Mathematics, vol. 254, Springer-Verlag, Berlin, 2009. MR 2464941
Michael Stoll, Galois groups over ${\bf Q}$ of some iterated polynomials, Arch. Math. (Basel) 59 (1992), no. 3, 239–244. MR 1174401, DOI https://doi.org/10.1007/BF01197321
Ioan Tomescu, Introduction to combinatorics, Collet’s Publishers Ltd., London, 1975. Translated from the Romanian by I. Tomescu and S. Rudeanu; Edited by E. Keith Lloyd. MR 0396275
B. L. van der Waerden, Die Zerlegungs-und Trägheitsgruppe als Permutationsgruppen, Math. Ann. 111 (1935), no. 1, 731–733 (German). MR 1513024, DOI https://doi.org/10.1007/BF01472249