Tropicalizing polynomial strata

Abstract

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