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Dynamical Galois groups of trinomials and Odoni’s conjecture. (English) Zbl 1462.11103

Summary: We prove that for every prime \(p\), there exists a degree \(p\) polynomial whose arboreal Galois representation is surjective, that is, whose iterates have Galois groups over \(\mathbb{Q}\) that are as large as possible subject to a natural constraint coming from iteration. This resolves in the case of prime degree a conjecture of Odoni from 1985. We also show that a standard height uniformity conjecture in arithmetic geometry implies the existence of such a polynomial in many degrees \(d\) which are not prime.

MSC:

11R32 Galois theory
37P15 Dynamical systems over global ground fields
14G05 Rational points
11D45 Counting solutions of Diophantine equations
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References:

[1] W.Aitken, F.Hajir and C.Maire, ‘Finitely ramified iterated extensions’, Int. Math. Res. Not.14 (2005) 855-880. · Zbl 1160.11356
[2] R.Benedetto and J.Juul, ‘Odoni’s conjecture for number fields’, Preprint, 2018, https://arxiv.org/abs/1803.01987. · Zbl 1470.11289
[3] C.Bombieri and W.Gubler, Heights in diophantine geometry (Cambridge University Press, Cambridge, 2006). · Zbl 1115.11034
[4] W.Bosma, J.Cannon and C.Playoust, ‘The Magma algebra system. I. The user language’, J. Symbolic Comput.24 (1997) 235-265. · Zbl 0898.68039
[5] S. D.Cohen, A.Movahhedi and A.Salinier, ‘Double transitivity of Galois groups of trinomials’, Acta Arithmetica82 (1997) 1-15. · Zbl 0893.11046
[6] S. D.Cohen, A.Movahhedi and A.Salinier, ‘Galois groups of trinomials’, J. Algebra222 (1999) 561-573. · Zbl 0939.12002
[7] B.Fein and M.Schacher, ‘Properties of iterates and composites of polynomials’, J. Lond. Math. Soc.54 (1996) 489-497. · Zbl 0865.12003
[8] A.Fröhlich and M.Taylor, Algebraic number theory (Cambridge University Press, New York, 1991).
[9] C.Gratton, K.Nguyen and T.Tucker, ‘ABC implies primitive prime divisors in arithmetic dynamics’, Bull. Lond. Math. Soc.45 (2013) 1194-1208. · Zbl 1291.37121
[10] D.Hilbert, ‘Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten’, J. reine angew. Math.110 (1892) 104-129. · JFM 24.0087.03
[11] W.Hindes, ‘Galois uniformity in arithmetic dynamics’, J. Number Theory148 (2015) 372-383. · Zbl 1391.37090
[12] S.Ih, ‘Height uniformity for algebraic points on curves’, Compos. Math.134 (2002) 35-57. · Zbl 1031.11041
[13] S.Ih, ‘Height uniformity for integral points on elliptic curves’, Trans. Amer. Math. Soc.358 (2006) 1657-1675. · Zbl 1118.14026
[14] R.Jones, ‘The density of prime divisors in the arithmetic dynamics of quadratic polynomials’, J. Lond. Math. Soc.78 (2008) 523-544. · Zbl 1193.37144
[15] R.Jones, ‘Galois representations from pre‐image trees: an arboreal survey’, Pub. Math. Besançon. (2013) 107-136. · Zbl 1307.11069
[16] R.Jones and M.Manes, ‘Galois theory of quadratic rational functions’, Comment. Math. Helv.89 (2014) 173-213. · Zbl 1316.11104
[17] C.Jordan, ‘Théorèmes sur les groupes primitifs’, J. de Math. Pures et Appl.16 (1871) 383-408. · JFM 03.0046.04
[18] J.Juul, ‘Iterates of generic polynomials and generic rational functions’, Preprint, 2014, http://arxiv.org/abs/1410.3814. · Zbl 1442.37120
[19] B.Kadets, ‘Large arboreal Galois representations’, Preprint, 2018, https://arxiv.org/pdf/1802.09074. · Zbl 1468.11228
[20] P.Llorente, E.Nart and N.Vila, ‘Discriminants of number fields defined by trinomials’, Acta Arith.43 (1984) 367-373. · Zbl 0493.12010
[21] A.Movahhedi and A.Salinier, ‘The primitivity of the Galois group of a trinomial’, J. Lond. Math. Soc.53 (1996) 433-440. · Zbl 0862.11063
[22] V.Nekrashevych, Self‐similar groups (American Mathematical Society, Providence, RI, 2005). · Zbl 1087.20032
[23] J.Neukirch, Algebraic number theory (Springer, Berlin, 1999). · Zbl 0956.11021
[24] R.Odoni, ‘On the prime divisors of the sequence \(w_{n + 1} = 1 + w_1 \cdots w_n\)’, J. Lond. Math. Soc.32 (1985) 1-11. · Zbl 0574.10020
[25] R.Odoni, ‘The Galois theory of iterates and composites of polynomials’, Proc. Lond. Math. Soc.51 (1985) 385-414. · Zbl 0622.12011
[26] H.Osada, ‘The Galois groups of the polynomials \(X^n + a X^l + b\)’, J. Number Theory25 (1987) 230-238. · Zbl 0608.12010
[27] J.Rotman, An introduction to the theory of groups, 4th edn, Graduate Texts in Mathematics (Springer, New York, 1995). · Zbl 0810.20001
[28] J.Smith, ‘General trinomials having symmetric Galois group’, Proc. Amer. Math. Soc.63 (1977) 208-212. · Zbl 0358.12012
[29] J.Specter, ‘Polynomials with surjective arboreal representation exist in every degree’, Preprint, 2018, https://arxiv.org/abs/1803.00434.
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