Critical point criteria and dynamically monogenic polynomials

Abstract Let K be a number field with ring of integers O K $\mathcal {O}_K$ script upper O Subscript upper K , and let f ( x ) ∈ O K [ x ] $f(x)\in \mathcal {O}_K[x]$ f left parenthesis x right parenthesis element of script upper O Subscript upper K Baseline left bracket x right bracket be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of f ( x ) $f(x)$ f left parenthesis x right parenthesis for the iterates of f ( x ) $f(x)$ f left parenthesis x right parenthesis to be monogenic polynomials. More generally, we give necessary and sufficient conditions for the backward orbits of elements of O K $\mathcal {O}_K$ script upper O Subscript upper K under f ( x ) $f(x)$ f left parenthesis x right parenthesis 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.