Monogenically Stable Polynomials

A monic polynomial f(x)∈ℤ[x] is called stable if fn(x) is irreducible over ℚ for all n≥1, where fn(x) denotes the nth iterate of f(x). Regardless of whether f(x) is irreducible over ℚ, if there exists some monic g(x)∈ℤ[x] such that g(fn(x)) is irreducible over ℚ for all n≥1, we say that f(x) is g-stable. Many authors have studied such polynomials since Odoni first introduced this concept of stability in 1985. We extend these concepts here by adding the additional restriction of monogeneity. A monic polynomial f(x)∈ℤ[x] is defined to be monogenic if f(x) is irreducible over ℚ and 1,θ,θ2,…,θdegf−1 is a basis for the ring of integers of ℚθ, where f(θ)=0. We say that f(x) is g-monogenically stable, if g(fn(x)) is monogenic for all n≥1, for some monic g(x)∈ℤ[x]. When g(x)=x, we simply say that f(x) is monogenically stable. In this article, we provide methods for constructing g-monogenically stable polynomials f(x), for various polynomials f(x) and g(x).