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.

The authors are grateful to the anonymous referee for many corrections and valuable suggestions that improved the manuscript. The authors would love to express their gratitude to Wade Hindes for reading an earlier draft of the manuscript and for several suggestions that helped the authors improve the manuscript. This work was initiated when M. O. Darwish was a master's student at Sabanci University under the supervision of M. Sadek. This work is supported by The Scientific and Technological Research Council of Turkiye, TUBITAK; research grant: ARDEB 1001/124F352.