Monodromy of stratified braid groups, II

The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical stratification. Tracking the positions of roots and critical points, there is a map from the fundamental group of a stratum into a braid group. We give a complete determination of this map. It turns out to be characterized by the geometry of the translation surface structure on \(\mathbb{C}\mathbb{P}^1\) $\mathbb{C}{\mathbb{P}}^1$ induced by the logarithmic derivative \(\textrm{d}f/f\) $\text{d}f/f$ of a polynomial in the stratum.