An algebraic approach to certain cases of Thurston rigidity (Q2845859)

Let \(\mathcal P_d^{\mathrm {crit}}\) be the space of monic centered degree-\(d\) polynomials with marked critical points \((c_1,\ldots, c_{d-1})\). Imposing natural conditions on the \(c_i\)'s, such as asking them to be preperiodic with specified periods and tail lengths, creates many subvarieties of \(\mathcal P_d^{\mathrm {crit}}\), and it follows from the very deep rigidity theorem of Thurston that these subvarieties often intersect transversally. This article gives a new algebraic proof of a special case of this result: the subvarieties NEWLINE\[NEWLINEC_{1,n} = \{(f,c_1,c_2): f^n(c_1) = c_1\} \qquad \text{ and } \qquad C_{2,m} = \{(f,c_1,c_2): f^m(c_2) = c_2\}NEWLINE\]NEWLINE in \(\mathcal P_3^{\mathrm {crit}}\) intersect transversally for any \(n,m\geq 1\).NEWLINENEWLINEIn fact, \textit{A. Epstein} [Bull. Lond. Math. Soc. 44, No. 1, 39--46 (2012; Zbl 1291.37057)] has now proved similar results more generally for \(\mathcal P_{d}^{\mathrm {crit}}\), where \(d\) is any prime power. As the author notes, the proof in the present paper is similar to the \(d=3\) case of Epstein's proof in that they both prove the \(3\)-adic integrality of intersection points and then prove the non-vanishing of the Jacobian determinant at intersection points by working modulo \(3\). But their proofs of integrality differ conceptually: Epstein's proof is more dynamical and involves a careful analysis of the \(3\)-adic valuations of the orbits of the critical points, while this proof is more algebraic and involves the degree estimates of the curves \(C_{1,n}\) and \(C_{2,m}\) as well as a resultant calculation. The author also uses the same methods to prove analogous results when one allows one or both of the \(c_i\)'s to be preperiodic with tail length \(1\).