Braid monodromy of univariate fewnomials

Let Cd⊂Cd+1 be the space of nonsingular, univariate polynomials of degree d. The Viète map V:Cd→Symd(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 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 Cd+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 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∕bZ with Bd∕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.