On the set of stable primes for postcritically infinite maps over number fields
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914092660293632 |
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| author | König, Joachim |
| author_facet | König, Joachim |
| contents | Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials $f$ (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of $f$ are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and in particular apply to $100\%$ of polynomials of any given odd degree, thus adding evidence to the conjecture that polynomials with a "large" set of stable primes are necessarily of a very specific shape, and in particular are necessarily postcritically finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13477 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the set of stable primes for postcritically infinite maps over number fields König, Joachim Number Theory Dynamical Systems Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials $f$ (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of $f$ are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and in particular apply to $100\%$ of polynomials of any given odd degree, thus adding evidence to the conjecture that polynomials with a "large" set of stable primes are necessarily of a very specific shape, and in particular are necessarily postcritically finite. |
| title | On the set of stable primes for postcritically infinite maps over number fields |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2408.13477 |