Eventual stability of pure polynomials over the rational field

Official URL: https://nyjm.albany.edu/j/2026/32-7v.pdf

A polynomial with rational coefficients is said to be pure with respect to a rational prime p if its Newton polygon has one slope. We establish the dynamical irreducibility, i.e., the irreducibility of all iterates, of a subfamily of pure polynomials, namely Dumas polynomials, with respect to a rational prime p under a mild condition on the degree. This provides iterative techniques to produce irreducible polynomials in ℚ[x] by composing pure polynomials of different degrees. In addition, for specific subfamilies of pure polynomials, we provide explicit bounds on the number of irreducible factors of the n-th iterate. These bounds are independent of n and improve upon existing results in the literature. During the course of this work, we characterize all polynomials whose degrees are large enough that are not pure, yet they possess pure iterates. This implies the existence of polynomials in ℤ[x] whose shifts are all dynamically irreducible.