The behavior under iteration of the critical points of polynomial maps playsan essential role in understanding its dynamics. We study the special casewhere the forward orbits of the critical points are finite. Thurston's theoremtells us that fixing a particular critical portrait and degree leads to onlyfinitely may possible polynomials (up to equivalence) and that, in many cases,their defining equations intersect transversely. We provide explicit algebraicformulae for the parameters where the critical points of the unicriticalpolynomials and bicritical cubic polynomials have a specified exact period. Wepay particular attention to the parameters where the critical orbits arestrictly preperiodic, called Misiurewicz points. Our main tool is thegeneralized dynatomic polynomial. We also study the discriminants of thesepolynomials to examine the failure of transversality in positive characteristicfor unicritical polynomials.
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