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Polynomial compositions with large monodromy groups and applications to arithmetic dynamics

Joachim König, Danny Neftin, Shai Rosenberg

Abstract

For a composition $f=f_1\circ\cdots \circ f_r$ of polynomials $f_i\in \mathbb Q[x]$ of degrees $d_i\geq 5$ with alternating or symmetric monodromy group, we show that the monodromy group of $f$ contains the iterated wreath product $A_{d_r}\wr \cdots\wr A_{d_1}$. A similar property holds more generally for polynomials that do not factor through $x^d$ or Chebyshev. We derive consequences to arithmetic dynamics regarding arboreal representations, and forward and backward orbits of such $f$. In particular, given an orbit $(a_n)_{n=0}^\infty$ of $f$ as above, we show that for "almost all" $a\in \mathbb Z$, the set of primes $p$ for which some $a_n$ is congruent to $a$ mod $p$ is "small".

Polynomial compositions with large monodromy groups and applications to arithmetic dynamics

Abstract

For a composition of polynomials of degrees with alternating or symmetric monodromy group, we show that the monodromy group of contains the iterated wreath product . A similar property holds more generally for polynomials that do not factor through or Chebyshev. We derive consequences to arithmetic dynamics regarding arboreal representations, and forward and backward orbits of such . In particular, given an orbit of as above, we show that for "almost all" , the set of primes for which some is congruent to mod is "small".
Paper Structure (15 sections, 30 theorems, 18 equations)

This paper contains 15 sections, 30 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maps, monodromy and ramification
  4. Imprimitive permutation groups
  5. Kernel properties
  6. Specialization
  7. Properties of decompositions that are necessary for largeness
  8. Invariant decompositions
  9. Nondiagonality
  10. Conjugation compatibility
  11. Main theorem and consequences
  12. Stable primes and prime divisors of dynamical sequences
  13. Largeness of arboreal representations
  14. Splitting
  15. A field version

Key Result

Theorem 1.1

Suppose $f=f_1\circ \cdots\circ f_r$ for $f_i\in {\mathbb{Q}}[x]$ of degree $d_i\geq 5$ with $\mathop{\mathrm{Mon}}\nolimits(f_i)\in\{A_{d_i},S_{d_i}\}$, $i=1,\ldots,r$. Then $\mathop{\mathrm{Mon}}\nolimits(f)$ contains a subgroup isomorphic to$A_{d_r}\wr \cdots \wr A_{d_1}$.

Theorems & Definitions (69)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 59 more