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Braid monodromy of univariate fewnomials

Alexander Esterov and Lionel Lang

Geometry & Topology 25 (2021) 3053–3077

Let 𝒞d d+1 be the space of nonsingular, univariate polynomials of degree d. The Viète map 𝒱 : 𝒞d Symd() 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 π1(𝒞d) and the Artin braid group Bd. For fewnomials, or equivalently for the intersection 𝒞 of 𝒞d with a collection of coordinate hyperplanes in d+1, the image of the map 𝒱: π1(𝒞) Bd is not known in general.

We show that the map 𝒱 is surjective provided that the support of the corresponding polynomials spans 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 b with Bdb. 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.

braid group, monodromy, fewnomial, tropical geometry
Mathematical Subject Classification 2010
Primary: 20F36, 55R80, 14T05
Received: 19 February 2020
Revised: 6 July 2020
Accepted: 6 August 2020
Published: 30 November 2021
Proposed: Mladen Bestvina
Seconded: Jim Bryan, Benson Farb
Alexander Esterov
Faculty of Mathematics
HSE University
Lionel Lang
Department of Electrical Engineering, Mathematics and Science
University of Gävle