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arXiv · 2412.10358

Critical Point Criteria and Dynamically Monogenic Polynomials

Joachim König·Hanson Smith·Zack Wolske
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Abstract

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.

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Joachim König, Hanson Smith, Zack Wolske. 2024-12-13. Critical Point Criteria and Dynamically Monogenic Polynomials. https://arxiv.org/abs/2412.10358

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