#### Volume 25, issue 6 (2021)

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Braid monodromy of univariate fewnomials

### Alexander Esterov and Lionel Lang

Geometry & Topology 25 (2021) 3053–3077
##### Abstract

Let ${\mathsc{𝒞}}_{d}\subset {ℂ}^{d+1}$ be the space of nonsingular, univariate polynomials of degree $d$. The Viète map $\mathsc{𝒱}:{\mathsc{𝒞}}_{d}\to {Sym}_{d}\left(ℂ\right)$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map ${\mathsc{𝒱}}_{\ast }$ at the level of fundamental groups realises an isomorphism between ${\pi }_{1}\left({\mathsc{𝒞}}_{d}\right)$ and the Artin braid group ${B}_{d}$. For fewnomials, or equivalently for the intersection $\mathsc{𝒞}$ of ${\mathsc{𝒞}}_{d}$ with a collection of coordinate hyperplanes in ${ℂ}^{d+1}$, the image of the map ${\mathsc{𝒱}}_{\ast }:{\pi }_{1}\left(\mathsc{𝒞}\right)\to {B}_{d}$ is not known in general.

We show that the map ${\mathsc{𝒱}}_{\ast }$ is surjective provided that the support of the corresponding polynomials spans $ℤ$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of ${\mathsc{𝒱}}_{\ast }$ is the expected wreath product of $ℤ∕bℤ$ with ${B}_{d∕b}$. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.

##### Keywords
braid group, monodromy, fewnomial, tropical geometry
##### Mathematical Subject Classification 2010
Primary: 20F36, 55R80, 14T05