Arithmetic properties of multiplier polynomials for certain polynomial maps

We investigate the arithmetic properties of the multiplier polynomials for certain $1$-parameter families of polynomials. In particular, we prove integrality theorems of multiplier polynomials for $z^d+c$, $(z-c)z^d + c$ and $z^{d+1}+cz$. As a corollary, we obtain the uniform upper bound of the naive height of parabolic parameters of unicritical polynomials. Moreover, we determined the quadratic parabolic parameters for $z^2 + c$. We also conditionally list parabolic parameters for $z^2 + c$ of fixed degrees.