Critical Point Criteria and Dynamically Monogenic Polynomials
Joachim König, Hanson Smith, Zack Wolske
TL;DR
This work develops a Critical Point Criterion that connects the critical points of a monic polynomial $f\in {\mathcal{O}}_K[x]$ to the monogenicity of iterates $f^n(x)-a$, framed via Uchida's criterion and the notion of ${\mathfrak{p}}$-maximality for primes ${\mathfrak{p}}$. The approach reduces monogenicity questions to ${\mathfrak{p}}$-adic valuations at lifts of $f'(x)$ modulo ${\mathfrak{p}}$, distinguishing non-exceptional and vanishing primes to handle simultaneous monogenicity across iterates. The authors apply this framework to unicritical polynomials, Chebyshev polynomials, and PCF compositions, deriving explicit prime-based conditions and density results that guarantee dynamical monogenicity and enable the construction of towers of monogenic number fields. By coupling PCF structure with rational critical values, the paper demonstrates positive-density families of parameters $a$ yielding dynamically monogenic pairs, thereby connecting arithmetic dynamics with classical monogenicity questions and providing practical criteria for generating monogenic extensions via backward orbits.
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.