  The behavior under iteration of the critical points of polynomial maps 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 portrait and degree leads to only
finitely may 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 the unicritical
polynomials and bicritical 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 positive characteristic
for unicritical polynomials.
