Tropicalizing polynomial strata

Abstract

Let ${\mathrm{Poly}}_D(\vec{\hat{I}}{¼}^{\ast })$ be the ramification stratum in the parameter space of degree $D\ge 2$ complex polynomials consisting of polynomials with ramification profile $\vec{\hat{I}}{¼}^{\ast }$. In this paper, we introduce a space ${\mathcal{T}}_D(\vec{\hat{I}}{¼}^{\ast })$ of framed decorated polynomial trees and identify it with the dynamical tropicalization of ${\mathrm{Poly}}_D(\vec{\hat{I}}{¼}^{\ast })$. A non-trivial point is that the tropicalization of ${\mathrm{Poly}}_D(\vec{\hat{I}}{¼}^{\ast })$ is not available a priori, as the stratum does not come with a previously known proper toroidal compactification. We resolve this issue by identifying ${\mathrm{Poly}}_D(\vec{\hat{I}}{¼}^{\ast })$ 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 ${\mathcal{T}}_D(\vec{\hat{I}}{¼}^{\ast })$ is isomorphic to this skeleton. We further show that the projectivized tree space $\mathbb{P}{\mathcal{T}}_D(\vec{\hat{I}}{¼}^{\ast })$ compactifies ${\mathrm{Poly}}_D(\vec{\hat{I}}{¼}^{\ast })$. Finally, we compare this compactification with the DeMarco--McMullen compactification of polynomial moduli by projectivized space of polynomial trees.