Misiurewicz points for polynomial maps and transversality

The behavior under iteration of the critical points of a polynomial map plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston’s theorem tells us that fixing a particular critical point portrait and degree leads to only finitely many possible polynomials (up to equivalence) and that, in many cases, their defining equations intersect transversely. We provide explicit algebraic formulae for the parameters where the critical points of all unicritical polynomials and of cubic polynomials have a specified exact period. We pay particular attention to the parameters where the critical orbits are strictly preperiodic, called Misiurewicz points. Our main tool is the generalized dynatomic polynomial. We also study the discriminants of these polynomials to examine the failure of transversality in characteristic p > 0 for the unicritical polynomials z + c.