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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.17872 |
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| _version_ | 1866929230694055936 |
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| author | König, Joachim Neftin, Danny Rosenberg, Shai |
| author_facet | König, Joachim Neftin, Danny Rosenberg, Shai |
| contents | 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". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17872 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial compositions with large monodromy groups and applications to arithmetic dynamics König, Joachim Neftin, Danny Rosenberg, Shai Number Theory 37P55, 11G99, 12F10, 20B05 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". |
| title | Polynomial compositions with large monodromy groups and applications to arithmetic dynamics |
| topic | Number Theory 37P55, 11G99, 12F10, 20B05 |
| url | https://arxiv.org/abs/2401.17872 |