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

We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial f (x) = c 0 + c 1 x d 1 + ⋯ + 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 split into 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 group over d k / d elements and Z / d Z . We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich. [ABSTRACT FROM AUTHOR]