Braid monodromy of univariate fewnomials

Abstract

Let ${\mathcal{C}}_d\subset {\mathbb{C}}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Viète map $\mathcal{V}:{\mathcal{C}}_d\to Sy{m}_d(\mathbb{C})$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map ${\mathcal{V}}_{\ast }$ at the level of fundamental groups realises an isomorphism between ${\pi }_1({\mathcal{C}}_d)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $\mathcal{C}$ of ${\mathcal{C}}_d$ with a collection of coordinate hyperplanes in ${\mathbb{C}}^{d+1}$, the image of the map ${\mathcal{V}}_{\ast }:{\pi }_1(\mathcal{C})\to B_d$ is not known in general. In the present paper, we show that the map ${\mathcal{V}}_{\ast }$ is surjective provided that the support of the corresponding polynomials spans $\mathbb{Z}$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of ${\mathcal{V}}_{\ast }$ is the expected wreath product of $\mathbb{Z}/b\mathbb{Z}$ 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.

18 pages, 7 figures. Version 3: we fixed a gap in the proof of Proposition 4.1

References in corpus (2)

• Galois theory for general systems of polynomial equations
• Monodromy of rational curves on toric surfaces

Cited by in corpus (1)

• Monodromy of rational curves on toric surfaces