Braid monodromy of univariate fewnomials

Esterov, Alexander; Lang, Lionel

doi:10.2140/gt.2021.25.3053

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Braid monodromy of univariate fewnomials

Alexander Esterov and Lionel Lang

Geometry & Topology 25 (2021) 3053–3077

DOI: 10.2140/gt.2021.25.3053

Abstract

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

We show that the map $𝒱_{*}$ 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 $𝒱_{*}$ is the expected wreath product of $ℤ ∕ b ℤ$ 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.

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Keywords

braid group, monodromy, fewnomial, tropical geometry

Mathematical Subject Classification 2010

Primary: 20F36, 55R80, 14T05

References

Publication

Received: 19 February 2020

Revised: 6 July 2020

Accepted: 6 August 2020

Published: 30 November 2021

Proposed: Mladen Bestvina

Seconded: Jim Bryan, Benson Farb

Authors

Alexander Esterov
Faculty of Mathematics HSE University Moscow Russia

Lionel Lang
Department of Electrical Engineering, Mathematics and Science University of Gävle Gävle Sweden

Volume 25, issue 6 (2021)