On PCF Polynomials

The author of [27] proves that the set of post-critically finite (PCF) polynomials of given degree is a set of bounded height, up to PGL_2-conjugacy. This result is extended to show that the set of monic polynomials g(z) with rational coefficients of given degree such that there exists a d ≥ 2 such that g(z^d) is PCF, is also a set of bounded height. Note that by fixing the degree of a polynomial and algebraic degree of its coefficients, the set of such PCF polynomials is in fact finite, and computable. Bounds on the coefficients for quartic PCF polynomials with rational coefficients are computed, and a search of the resulting space yields 16 distinct conjugacy classes. Infinite families of PCF polynomials containing each of these distinct conjugacy classes are found, giving a lower bound on the number of such conjugacy classes in terms of degree d.