  Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular,
univariate polynomials of degree $d$. The Vi\`{e}te map $\mathscr{V} :
\mathcal{C}_d \rightarrow Sym_d(\mathbb{C})$ sends a polynomial to its
unordered set of roots. It is a classical fact that the induced map
$\mathscr{V}_*$ 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 $\mathscr{V} _* : \pi_1(\mathcal{C}) \rightarrow B_d$ is not known in
general. In the present paper, we show that the map $\mathscr{V} _*$ 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 $\mathscr{V} _*$ 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.
