# zbMATH — the first resource for mathematics

Large arboreal Galois representations. (English) Zbl 07161186
Summary: Given a field $$K$$, a polynomial $$f \in K [x]$$ of degree $$d$$, and a suitable element $$t \in K$$, the set of preimages of $$t$$ under the iterates $$f^{\circ n}$$ 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 $$d \geq 20$$ is even and $$K = \mathbb{Q}$$ 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 $$f \in F [x]$$ in which the corresponding Galois groups are the monodromy groups of the ramified covers $$f^{\circ n} : \mathbb{P}_F^1 \rightarrow \mathbb{P}_F^1$$.

##### MSC:
 11R32 Galois theory 11F80 Galois representations 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
Full Text:
##### References:
 [1] Benedetto, Robert L.; Juul, Jamie, Odoni’s conjecture for number fields, Bull. Lond. Math. Soc., 51, 2, 237-250 (2019), MR3937585 · Zbl 07094878 [2] Dixon, John D.; Mortimer, Brian, Permutation Groups, Graduate Texts in Math., vol. 163 (1996), Springer-Verlag: Springer-Verlag New York-Heidelberg-Berlin, MR1409812 (98m:20003) · Zbl 0951.20001 [3] Guralnick, Robert M.; Saxl, Jan, Monodromy groups of polynomials, (Groups of Lie Type and Their Geometries. Groups of Lie Type and Their Geometries, Como (1995), Cambridge University Press), 125-150, MR1320519 (96b:20003) · Zbl 0848.12002 [4] Jones, Rafe, Galois representations from pre-image trees: an arboreal survey, (Actes de la Conféren“Théorie des Nombres et Applicati”. Actes de la Confére“Théorie des Nombres et Applicat”, Publ. Math. Besançon Algèbre Théorie Nr. (2013), Presses Univ. Franche-Comté: Presses Univ. Franche-Comté Besançon), 107-136, English, with English and French summaries, MR3220023 · Zbl 1307.11069 [5] Juul, Jamie, Iterates of generic polynomials and generic rational functions, Trans. Amer. Math. Soc., 371, 2, 809-831 (2019), MR3885162 · Zbl 1442.37120 [6] Lang, Serge, Algebra, Graduate Texts in Mathematics, vol. 211 (2002), Springer-Verlag: Springer-Verlag New York, MR1878556 (2003e:00003) · Zbl 0984.00001 [7] Looper, Nicole R., Dynamical Galois groups of trinomials and Odoni’s conjecture, Bull. Lond. Math. Soc., 51, 2, 278-292 (2019), MR3953120 · Zbl 1462.11103 [8] Müller, Peter, Primitive monodromy groups of polynomials, Contemp. Math., 186, 385-401 (1995), MR1352284 (96m:20004) · Zbl 0840.12001 [9] Nagura, Jitsuro, On the interval containing at least one prime number, Proc. Japan Acad., 28, 4, 177-181 (1952), MR050615 · Zbl 0047.04405 [10] Nekrashevych, Volodymyr, Iterated monodromy groups, (Groups St Andrews 2009 in Bath. Volume 1. Groups St Andrews 2009 in Bath. Volume 1, London Math. Soc. Lecture Note Ser., vol. 387 (2011)), 41-93, MR2858850 · Zbl 1235.37016 [11] Odoni, R. W.K., The Galois theory of iterates and composites of polynomials, Proc. Lond. Math. Soc. (3), 51, 3, 385-414 (1985), MR805714 (87c:12005) · Zbl 0622.12011 [12] Pink, Richard, Profinite iterated monodromy groups arising from quadratic polynomials (2013-09-24), Preprint [13] Specter, Joel, Polynomials with surjective arboreal Galois representation exist in every degree (2018), Preprint, available at [14] Stoll, Michael, Galois groups over Q of some iterated polynomials, Arch. Math. (Basel), 59, 3, 239-244 (1992), MR1174401 (93h:12004) · Zbl 0758.11045 [15] Turnwald, Gerhard, On Schur’s conjecture, J. Aust. Math. Soc. A, 58, 3, 312-357 (1995), MR1329867 (96a:11135) · Zbl 0834.11052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.