2026/07/16 by Yan Mary He, Chenxi Wu
#math.AG #math.CV #math.DS
Let \rm PolyD(μ^*) be the ramification stratum in the parameter space of degree D ≥ 2 complex polynomials consisting of polynomials with ramification profile μ^*. In this paper, we introduce a space TD(μ^*) of framed decorated polynomial trees and identify it with the dynamical tropicalization of \rm PolyD(μ^*). A non-trivial point is that the tropicalization of \rm PolyD(μ^*) is not available a priori, as the stratum does not come with a previously known proper toroidal compactification. We resolve this issue by identifying \rm PolyD(μ^*) with a rigidified framed Hurwitz space. Using the twisted admissible cover compactification, together with additional rigidifying data, we construct a proper toroidal compactification and hence an associated Berkovich skeleton. We prove that TD(μ^*) is isomorphic to this skeleton. We further show that the projectivized tree space \mathbb PTD(μ^*) compactifies \rm PolyD(μ^*). Finally, we compare this compactification with the DeMarco--McMullen compactification of polynomial moduli by projectivized space of polynomial trees.