Stretching Newton polygons using pure polynomials

Fix a prime p and polynomials f, g Qₚx. If the Newton polygon of g consists of a single segment with negative slope, we show under some mild conditions that the Newton polygon of f g is identical to that of f, but stretched horizontally by a factor of deg g. Specializing to the case f=g, this implies that all iterates of certain pure polynomials are irreducible, recovering a classical result of Odoni on the dynamical irreducibility of Eisenstein polynomials. We also prove that the Taylor polynomials of the exponential function (a well-known family of irreducible polynomials) remain irreducible upon composition with all iterates of certain pure polynomials of large enough degree.