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Braid monodromy of univariate fewnomials
Let C_d be the space of non-singular, univariate polynomials of degree d. The Viète map V 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, the image of the map V_∗:π_1(C)→B_d is not known in general. In the present paper, 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 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.