A Dynamical Variant Of The Andre-Oort Conjecture

In the moduli space MPd of degree d polynomials, special subvarieties are those cut out by critical orbit relations, and then special points are the post-critically finite polynomials. It was conjectured that in MPd, subvarieties containing a Zariski-dense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case d = 3.