Critical Point Criteria and Dynamically Monogenic Polynomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910832415211520 |
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| author | König, Joachim Smith, Hanson Wolske, Zack |
| author_facet | König, Joachim Smith, Hanson Wolske, Zack |
| contents | Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $f(x)\in\mathcal{O}_K[x]$ be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of $f(x)$ for the iterates of $f(x)$ to be monogenic polynomials. More generally, we give necessary and sufficient conditions for the backwards orbits of elements of $\mathcal{O}_K$ under $f(x)$ to be monogenerators. We apply our criteria to construct novel examples of dynamically monogenic polynomials, yielding infinite towers of monogenic number fields with the backward orbit of one monogenerator giving a monogenerator at the next level. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical Point Criteria and Dynamically Monogenic Polynomials König, Joachim Smith, Hanson Wolske, Zack Number Theory 11R04, 11R21, 37P05 Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $f(x)\in\mathcal{O}_K[x]$ be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of $f(x)$ for the iterates of $f(x)$ to be monogenic polynomials. More generally, we give necessary and sufficient conditions for the backwards orbits of elements of $\mathcal{O}_K$ under $f(x)$ to be monogenerators. We apply our criteria to construct novel examples of dynamically monogenic polynomials, yielding infinite towers of monogenic number fields with the backward orbit of one monogenerator giving a monogenerator at the next level. |
| title | Critical Point Criteria and Dynamically Monogenic Polynomials |
| topic | Number Theory 11R04, 11R21, 37P05 |
| url | https://arxiv.org/abs/2412.10358 |