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We introduce a new technique to prove connectivity of subsets of coveringspaces (so called inductive connectivity), and apply it to Galois theory ofproblems of enumerative geometry. As a model example, consider the problem ofpermuting the roots of a complex polynomial $f(x) = c_0 + c_1 x^{d_1} + \ldots+ c_k x^{d_k}$ by varying its coefficients. If the GCD of the exponents is $d$,then the polynomial admits the change of variable $y=x^d$, and its roots splitinto necklaces of length $d$. At best we can expect to permute these necklaces,i.e. the Galois group of $f$ equals the wreath product of the symmetric groupover $d_k/d$ elements and $\mathbb{Z}/d\mathbb{Z}$. The aim of this paper is toprove this equality and study its multidimensional generalization: we show thatthe Galois group of a general system of polynomial equations equals theexpected wreath product for a large class of systems, but in general thisexpected equality fails, making the problem of describing such Galois groupsunexpectedly rich.

Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products