Arithmetic properties of multiplier polynomials for certain polynomial maps

We investigate the arithmetic properties of the multiplier polynomials for certain [Formula: see text]-parameter families of polynomials. In particular, we prove integrality theorems about the multiplier polynomials for [Formula: see text], [Formula: see text] and [Formula: see text]. As a corollary, we obtain a uniform upper bound on the naive height of parabolic parameters for unicritical polynomials. Moreover, we determine the quadratic parabolic parameters for [Formula: see text]. We also conditionally list the parabolic parameters for [Formula: see text] of certain fixed degrees.